On the geometry of diffusion operators and stochastic flows

K. D. Elworthy  Y. Le Jan  X.-M. Li

June 1, 1998

Contents

1 Construction of connections 7 1.1 Construction of connections . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Basic Classes of Examples . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3 Adjoint connections, torsion skew symmetry, basic formulae . . . . . . 18 1.4 Example: Homogeneous spaces continued . . . . . . . . . . . . . . . . . 25

2 The infinitesimal generators and associated operators 29 2.1 The irrelevance of drift in dimension greater than 1 . . . . . . . . . . . 29 2.2 Torsion Skew Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.3 The ‘divergence operator’ δ̂ . . . . . . . . . . . . . . . . . . . . . . . . 36 2.4 Hörmander form generators on differential forms . . . . . . . . . . . . . 40 2.5 On the infinitesimal generator . . . . . . . . . . . . . . . . . . . . . . . 46

2.5.1 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5.2 Symmetricity of the generator A^{q} . . . . . . . . . . . . . . . . . 47

3 Decomposition of noise and filtering 55

3.1 A canonical decomposition of the noise driving a stochastic differential

equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.2 Canonical decomposition of the Gaussian field W_{t} . . . . . . . . . . . . 58 3.3 Filtering out redundant noise . . . . . . . . . . . . . . . . . . . . . . . 61

3.3.1 When A does not belong to the image of X . . . . . . . . . . . 67 3.3.2 The inverse derivative flow . . . . . . . . . . . . . . . . . . . . . 69 3.3.3 Integrability of certain C^{r} norms for compact M . . . . . . . . . 69 3.3.4 The semigroup on forms: Bochner type vanishing theorems … 71 3.3.5 Bismut formulae . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4 Application: Analysis on spaces of paths 74 4.1 Integration by parts and Clark-Ocone formulae . . . . . . . . . . . . . 76 4.2 Logarithmic Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . 80 4.3 Analysis on C_{id}(DiffM) . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

5 Stability of stochastic dynamical systems 85

6 Appendices 92 A Universal Connections as L-W connection . . . . . . . . . . . . . . . . 92 B Creation and Annihilation operators (notation for section 2.4) . . . . . 97 C Basic formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 D List of notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Introduction

The concepts of second order semi-elliptic operator, Markov semi-group, diffusion process, diffusion measures on path spaces essentially give different pictures of the same fundamental objects, with related Riemannian or sub-Riemannian geometry. Here we consider a different layer of structure centred around the concepts of sums of squares of vector fields, stochastic differential equations, stochastic flows and Gaussian vector fields; again essentially equivalent, and this time with associated metric linear connections on tangent bundles and subbundles of tangent bundles. The difference between these two levels of structure can be seen from the fact that if a semi-elliptic differential operator on functions on a manifold M is given a representation as a sum of square of vector fields (“Hörmander form”) it automatically gets an extension to an operator on differential forms. In exactly the same way representing a diffusion process as the one point motion of a stochastic flow determines a semi-group acting on differential forms (by pulling the form back by the flow and taking expectation.) Given a regularity condition there is an associated linear connection and adjoint ‘semi-connection’ in terms of which these operators can be simply described (e.g. by a Weitzenbock formula) as can many other important quantities (e.g. existence of moment exponents for stochastic flows). Moreover in the stochastic picture the connections remain relevant in the collapse from this level to the simpler one giving new results and new proofs of results e.g. on path space measures.

In more detail: Chapter 1 is connected with the construction of linear connections of vector bundles as push forwards of connections on trivial bundles. This is a direct analogue of the classical and elementary construction of the covariant derivative of a vector field on a submanifold of Euclidean space, leading to the Levi-Civita connections (Example 1B). Narasimhan & Ramadan’s theorem of universal connections is evoked to assure us that all metric connections can be obtained this way (Theorem 1.1.2). We then go on to consider the various forms in which this construction will appear in situations described above. (E.g. how certain Gaussian fields of sections determine a connection.) Homogeneous spaces give a good class of examples described in some detail in §1.1 B. The notion of adjoint connection or semi-connection on a subbundle E of the tangent bundle T M to our underlying manifold M is described in §1.3. A semi-connection allows us to differentiate vector fields on M in E-directions. They play an important role in the theory. One difficulty is that the adjoint of a metric connection may not be metric for any metric (Corollary 1.3.7). In general Hörmander type hypoellipticity conditions on our generator A, or equivalently on E, play little role in this article. However in Theorem 1.3.9 we show how they are related to the behaviour of parallel translations with respect to associated semi-connections.

In chapter 2 we concentrate on a generator A given in Hörmander form, and its associated stochastic differential equation (s.d.e.). A first result is Theorem 2.1.1 which shows in particular that (for dimM > 1) any elliptic diffusion operator can be written as a sum of squares with no first order term, or equivalently that any elliptic diffusion is given by a Stratonovich equation with no drift term. The extension A^{q} of A to q-forms is shown to have the form A^{q} = −(dδ̂ + δ̂d) for a certain operator δ̂ from q-forms to q − 1 forms (Proposition 2.3.1) and also a Weitzenbock form A^{q} = \frac{1}{2}trace ˆ∇^{2} − \frac{1}{2}R^{q} (if there is no drift term A) (Theorem 2.4.2). Driver’s notion of torsion skew symmetry is investigated in §2.2 in order to discuss the operators δ̂, and when they are L^{2} adjoints of the exterior derivative d, in §2.3. Later, §3.3.3, the semigroups associated to these operators are used to obtain Böchner type vanishing theorems under positivity conditions on R^{q}.

The question of the symmetricity of A^{q} with respect to some measure on M is discussed in §2.5.2. Theorem 2.5.1 gives a fairly definitive result for A^{q} with the zero order terms removed. However conditions under which R^{q} is symmetric seem not so easy to find if q > 1. For q = 1 this reduces to symmetricity of the Ricci curvature R̆ic which is shown in Proposition C.6 of the Appendix to hold in the torsion skew symmetric case if and only if the torsion tensor T̆ determines a coclosed differential 3-form, c.f. [Dri92]. These sections are not used later in this article.

The main applications in stochastic analysis start with Chapter 3. The basic idea is that the diffusion coefficient of an s.d.e often has a kernel: so that there is “redundant noise” from the point of view of the one point motion. We extend the results from the gradient case in [EY93] to our more general, possibly degenerate, situation giving a canonical decomposition of the noise into its redundant and non-redundant parts. We then show how this can be used to filter out the redundant noise in general situations. (This filtering out corresponds to the collapse in levels of structure mentioned above.) On the way we have to discuss conditional expectations of vector fields along the sample paths of our process, Definition 3.3.2. All this is done in some generality, e.g. allowing for the possibility of explosion. The main application is to the derivative process T ξ_{t} of a stochastic flow: Theorem 3.3.7 and Theorem 3.3.8. When the redundant noise is filtered out the process becomes a “damped’ or Dohrn-Guerra type parallel translation using the associated semi-connection. This procedure works equally for the derivative of the Itô map ω ↦ ξ_{t}(x_{0})(ω) in the sense of Malliavin Calculus from which follow integration by parts theorems for possibly degenerate diffusion measures, Theorem 4.1.1. For gradient systems, using [EY93], this method was used by [EL96] and was suggested by [AE95]. The Levi-Civita connection appears in that case (which is why gradient systems behave so nicely), but in the degenerate case which is allowed here the connections are on arbitrary subbundles of T M and there is no unique particularly well behaved connection to use. Hypoellipticity is not assumed. The “admissible” vector fields are those which satisfy a natural “horizontality” condition, §4.1 B and §4.1 C. Closely related is a Clark-Ocone formula (Theorem 4.1.2) expressing suitably smooth functions on path space as stochastic integrals with respect to the predictable projection of their gradient. From this we use the method given in [CHL] to obtain a Logarithmic Sobolev inequality for our diffusion measures, Theorem 4.2.1. Our “damping” of the parallel translation means that no curvature constants appear: indeed since in general we have no Riemannian metric given on M it would be unnatural to have such constants. Logarithmic Sobolev inequalities automatically imply spectral gap inequalities and the constancy of functionals with vanishing gradient (or equivalently whose derivatives vanish on admissible vector fields), Corollary 4.1.3: a non-trivial result even for Frechet smooth functions on path space for the case of degenerate diffusions. In Theorem 4.1.1 the corresponding results are proved for the measures on paths on the diffeomorphism group DiffM of M coming from stochastic flows, or equivalently from Wiener processes on DiffM [Bax84].

Chapter 5 is concerned with applications to stability properties of stochastic flows. In particular upper and lower bounds for moment exponents are obtained in terms of the Weitzenbock curvatures of the associated connection and a generalization of the second fundamental form to our situations: Theorem 5.0.5. This gives a criterion for moment stability in terms of ‘stochastic positivity’ of a certain expression in the quantities with consequent topological implications: Corollary 5.0.6.

A weakness of these results is that we usually require the adjoint semi-connection to be metric for some metric. Theorem 5.0.7 shows that the lack of this condition really is reflected in the behaviour of the flow.

Chapter 6 consists of technical appendices. The first gives a detailed description of how the push forward construction of connections we use relates to Narasimhan & Ramanan’s pull back of the universal connections. This is needed in the proof of Theorem 1.1.2. The other appendices give the notation of annihilation and creation operators used in the discussion of the Weitzenbock curvatures in section 2.4 and some basic formulae and curvature calculations for connections given in the L-W form.

The connection determined by a non-degenerate stochastic flow first appeared in [LJW84]: for this reason we have called it the LeJan-Watanabe or L-W connection. It was also discovered in the context of quantum flows in [AA96] and for sums of squares of vector fields in [PVB96]. It is used for analysis on loop spaces in [Aid96]. For the non-degenerate case many of the results given here were described in [ELJL97a] with announcements for degenerate situations in [ELJL97b]. They were stimulated by [EY93]. The Chentsov-Amari α-connections in statistics are rather different. They are in general non-metric if α ≠ 0 and torsion free, see [Ama85], pp42, 46.

Acknowledgement: The authors would like to thank MSRI, Institut Henri Poincaré and Universitaet Bochum (as well as their home institutes) for hospitality during the completion of this project. Support is acknowledged from EU grant ERBF MRX CT 960075 A, NSF grant DMS-9626142, Alexander Humboldt-Stiftung, and the British Council.

K. D. ELWORTHY, MATHEMATICS INSTITUTE, UNIVERSITY OF WARWICK, COVENTRY CV4 7AL, UK Y. LE JAN, DÉPARTMENT DE MATHÉMATIQUE, UNIVERSITÉ PARIS SUD, 91405 ORSAY, FRANCE

XUE-MEI LI, DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CONNECTICUT, 196 AUDITORIUM ROAD, STORRS, CT 06269 USA. e-mail address: xmli@math.uconn.edu

Chapter 1

Construction of connections

We consider connections on a C^{∞} vector bundle E over a smooth manifold M deter-

mined by a split surjection of vector bundles \underline{H} \underrightarrow{Y_{⟵}} E ⟶ 0 where \underline{H} = M × H is the trivial bundle with fibre a Hilbertable space H. The characterization of such a connection ˘∇ is that for each x ∈ M

˘∇_{v} (X(·)e) ≡ 0 all v ∈ T_{x}M and e ∈ ImageY (x) .

When M and E are finite dimensional and E has a Riemannian metric all metric connection on E can be obtained this way for some finite dimensional H. These connections can also be considered to be induced by Gaussian measures on the space of C^{∞} sections of E. In §1.2 and §1.4 some basic examples are given. They describe the connections arising from certain Gaussian fields, operators in Hörmander form, stochastic differential equations, and homogeneous space structures.

For E a subbundle of T M there is also an adjoint ’semi-connection’ ˆ∇, investigated in §1.3. In particular we show ˆ∇ is metric with respect to some Riemannian metric on M if and only if for one set of x_{0}, y_{0} ∈ M and T > 0 the parallel translation ˆ//_{t} using ˆ∇ along {ξ_{t}^{T,y_{0}}(x_{0})}, the solution ξ_{t}(x_{0}) to the stochastic differential equation dx_{t} = X(x_{t}) ∘ dB_{t} conditioned to satisfy ξ_{T} (x_{0}) = y_{0}, is a bounded L(T_{x_{0}}M, T_{y_{0}}M)- valued process.

1.1 Construction of connections

A. Consider a C^{∞} manifold M, a C^{∞} vector bundle π : E → M over M and a C^{∞} vector bundle homorphism X : \underline{H} → E of a trivial bundle \underline{H} = M × H, where H is a Hilbertable space. We will consider only real bundles (and manifolds) here. At this stage M, E, H could be infinite dimensional (but separable, with M metrizable); however our main focus will be on cases with M and E finite dimensional. In this situation we shall write n = dim M, p = fibre dimension of E, with m = dim H if dim H < ∞.

Suppose X is surjective and Y a chosen right inverse to X

\underline{H} \underrightarrow{Y_{⟵}} E ⟶ 0.

Our situation is very similar to a special case of that of Harvey and Lawson in [HL93]. Let Γ(E) denote the space of smooth sections of E, and set E_{x} = π^{−1}(x), x ∈ M. Write X(x) = X(x, ·) : H → E_{x}. For u in E let Z^{u} be the section given by

Z^{u}(x) = X(x)Y (π(u))u.  (1.1.1)

Proposition 1.1.1 There is a unique linear connection ˘∇ on E such that for all u_{0} ∈ E_{x_{0}}, x_{0} ∈ M the covariant derivative of Z^{u_{0}} vanishes at x_{0}. It is the ’push forward’ connection defined by

˘∇_{v_{0}}Z = X(x_{0})d (Y (·)Z(·)) (v_{0}), v_{0} ∈ T_{x_{0}}M, Z ∈ Γ(E)  (1.1.2)

where d refers to the usual derivative d (Y (·)Z(·)) : T M → H of the map Y (·)Z(·) : M → H.

Proof. Certainly (1.1.2) defines a covariant differentiation. Let ˜∇ be any linear connection on E. We have

Z(·) = X(·)Y (·)Z(·)

whence, for v ∈ T_{x_{0}}M,

\begin{aligned}˜∇_{v}Z & = X(x_{0})d (Y (·)Z(·)) (v) + ˜∇_{v} [X(·) ( Y (x_{0})Z(x_{0}) )] \\ & = ˘∇_{v}Z + ˜∇_{v}Z^{Z(x_{0})}. \end{aligned}  (1.1.3)

Since ˜∇ is assumed to be a genuine connection (not just a covariant differentiation: a point only relevant if E is infinite dimensional) and since also the map

T M × E → E
(v, u) ↦ ˜∇_{v}Z^{u}

gives a smooth section of the bundle of bilinear maps L(T M, E; E) we see that ˘∇ is a smooth connection on E, (e.g. [Eli67]). Taking ˜∇ = ˘∇ in (1.1.3) we see that ˘∇ has the property required. Uniqueness also follows from (1.1.3). //

B. We shall be mainly interested in metric connections. These will arise in two, essentially equivalent, forms which we will call the metric form and the Gaussian form respectively. However the examples coming from homogeneous spaces are more easily understood in the more general non-metric framework and these will also be described below, in §1.2.

In the “metric” form H is now a Hilbert space, inner product ⟨, ⟩ ≡ ⟨, ⟩_{H} and so the surjective homomorphism X induces a Riemannian metric {⟨, ⟩_{x} : x ∈ M} on E. The right inverse Y is chosen to be the adjoint of X, Y = X^{∗}.

Theorem 1.1.2 Let H be a Hilbert space, and Y the adjoint of X with respect to the induced metric on E by X. Then the connection ˘∇ is adapted to the Riemannian metric. Moreover if M and E are finite dimensional any metric connection for any Riemannian metric on E can be obtained this way from some such X with H some finite dimensional Hilbert space.

Proof. Take a vector field U and a vector v ∈ T_{x_{0}}M. Then

d ⟨U, U⟩ (v) = 2⟨d (Y (·)U(·)) (v), Y (x_{0})U(x_{0})⟩
= 2⟨X(x_{0})(d (Y (·)U(·)) (v), U(x_{0})⟩ = 2⟨ ˘∇_{v}U, U⟩.

This shows that ˘∇ is metric. The fact that any metric connection arises this way in the finite dimensional situation comes from Narasimhan and Ramanan’s theorem [NR61] on universal connections. In the finite dimensional case the connection ˘∇ is precisely the pull back of the universal connection over the Grassmanian G(m, p) of p-planes in H by the map x ↦ [ image of Y (x) : E_{x} → H]; for details see §A in the Appendix. Narasimhan and Ramanan show that any metric connection can be obtained as such a pull back.

In this situation we shall call ˘∇ the LeJan-Watanabe or L-W, connection determined by X, or by (X, ⟨, ⟩), for reasons described at the end of §1.3B below.

Example 1B (Gradient systems). Let j : M → ℝ^{m} be an immersion. Define X(x) : ℝ^{m} → T_{x}M to be the orthogonal projection of ℝ^{m} on T_{x}M, identified with its image under the differential dj of j, so that X(x)e = grad⟨X(·), e⟩_{ℝ^{m}} using the induced metric on M. Then Y (x) : T_{x}M → ℝ^{m} is the inclusion, T j, and we have the classical construction of the Levi-Civita connection for this metric.(That it has no torsion can also be seen from the formula (2.2.3) below.)

C. For the ‘Gaussian form’ suppose we have a mean zero Gaussian field W of sections of E. In its most general form W would be a section of the pull back p^{∗}E of Γ(E) over the projection p : Ω×M → M where (Ω, F, P) is a probability space. Thus W_{x}(ω) := W (ω, x) ∈ E_{x} for each x ∈ M, ω ∈ Ω. We will assume that W (ω, ·) is C^{∞} for each ω ∈ Ω. The more concrete manifestation comes from a Gaussian measure γ on some subspace of the C^{∞} sections of E. Then (Ω, F, P) = (Γ(E), F, γ), the canonical space, for F the σ-algebra of cylindrical subsets of Γ(E), and we identify W_{x} with the evaluation map ρ_{x} : Γ(E) → E_{x}, given by ρ_{x}(U) = U(x). See [Bax76]. For any suitable function f on Γ(E) we write 𝔼f or 𝔼f(W) for ∫_{Ω} f(W (ω, ·))P (dω) (equivalently ∫_{Γ(E)} f(U)dγ(U) in the canonical picture). Let γ_{x} be the law of W_{x}, a Gaussian measure on E_{x}. We make the nondegeneracy assumption that each γ_{x} is non-degenerate and so in the finite dimensional case has the form

γ_{x}(B) = (2π)^{−p/2} ∫_{B} e^{−⟨y,y⟩_{x}^{2}}dy

for some ⟨, ⟩_{x} on E_{x}.

Proposition 1.1.3 There is a unique connection ∇^{γ} on E such that the random variable ∇_{v}^{γ}W is independent of W (x_{0}) for any v ∈ T_{x_{0}}M, x_{0} ∈ M. It is given in terms of the conditional expectation by

∇_{v}^{γ}Z = \frac{d}{dt} 𝔼 \{W (x_{0}) |W (σ(t)) = Z(σ(t))\} |_{t=0};  (1.1.4)

or equivalently

∇_{v}^{γ}Z = \frac{d}{dt}𝔼W (x_{0}) ⟨W (σ(t)), Z(σ(t))⟩_{σ(t)}|_{t=0}  (1.1.5)

for any C^{1} curve σ with σ̇(0) = v, v ∈ T_{x_{0}}M, and is adapted to the metric {⟨, ⟩_{x}, x ∈ M}. Moreover

(i) Let H_{γ} be the reproducing kernel Hilbert space of γ, then ∇^{γ} is the L-W connection for (X, ⟨, ⟩_{H_{γ}}) where X(x, h) = ρ_{x}(h) = h(x). (ii) If E is a finite dimensional vector bundle over a finite dimensional M every metric connection can be considered as ∇^{γ} , given by (1.1.4), for some Gaussian measure γ on Γ(E) with finite dimensional support.

Proof. Recall that the reproducing kernel Hilbert space H_{γ} of γ is the same as the Cameron-Martin space H of γ and is a Hilbert space, here necessarily consisting of C^{∞} functions. Among its standard properties are:

(i) The restriction of ρ_{x} to H maps onto E_{x}, each x ∈ M and induces the inner product ⟨, ⟩_{x}. It will also be denoted by ρ_{x}.

(ii) The reproducing kernel k, a section of the vector bundle

∪_{x∈M,y∈M}L(E_{x}, E_{y}) → M × M,

defined by the reproducing property that k(x, ·)(v) belongs to H each v ∈ E_{x} and for all h ∈ H,

⟨k(x, ·)v, h⟩_{H_{γ}} = ⟨h(x), v⟩_{x},  (1.1.6)

is also the covariance of γ:

k(x, y)v = 𝔼⟨W (x), v⟩_{x}W (y), v ∈ E_{x}, x, y ∈ M.  (1.1.7)

See [Bax76]. From (1.1.7) we see

k(x, y)W (x) = 𝔼 {W (y)|W (x)} ∈ T_{y}M  (1.1.8)

and so

k(x, ·)v = 𝔼 {W (·) | W (x) = v } ∈ H ⊂ Γ(E)  (1.1.9)

for all v ∈ E_{x}, and x, y ∈ M.

From this we see that the defining equation (1.1.4) or (1.1.5) for ∇_{v}^{γ}Z can be written

∇_{v}^{γ}Z = \frac{d}{dt}k(σ(t), x_{0})Z(σ(t)) |_{t=0}  (1.1.10)
= \frac{d}{dt}𝔼W (x_{0}) ⟨W (σ(t)), Z(σ(t))⟩_{σ(t)}|_{t=0} .  (1.1.11)

If we set X(x) = ρ_{x} so X : \underline{H} → E we see from (1.1.6) that the adjoint map Y to X, using the induced Riemannian metric is just k:

Y (x) = ρ_{x}^{∗} = k(x, ·) : E_{x} → H
k(x, y)(·) = X(y)Y (x) : E_{x} → E_{y}.

We are therefore in the ’metric’ form discussed in Theorem 1.1.2 and ˜∇ is just the L-W connection ˘∇ of ρ, ⟨, ⟩_{H_{γ}}.

For the characterization in terms of the independence of ˜∇_{Z}W and W observe that for u ∈ E_{x_{0}} the reproducing vector field Z^{x_{0},u}, or Z^{u} as in (1.1.1) in the metric form, is given by

Z^{u}(y) = X(y)Y (x_{0})u = k(x_{0}, y)u  (1.1.12)
= 𝔼 {W (y) | W (x_{0}) = u}  (1.1.13)

so that for any linear connection ˜∇ on E we have

˜∇_{v}Z^{u} = 𝔼 { ˜∇_{v}W | W (x_{0}) = u} .

Thus if v ∈ T_{x_{0}}M we have ˜∇_{v}Z^{u} = 0 for all u if and only if

𝔼⟨ ˜∇_{v}W, W (x_{0})⟩_{x_{0}} = 0

since ˜∇_{v}W and W (x_{0}) are E_{x_{0}}-valued Gaussian random variables (this is exactly the condition for the independence of W (x_{0}) and ˜∇_{v}W). By Proposition 1.1.1 this proves uniqueness, i.e. that ˜∇_{v}W and W (x_{0}) are independent for all x_{0} and v ∈ T_{x_{0}}M implies ˜∇ = ∇^{γ}. It also shows that

𝔼⟨∇_{v}^{γ}W, W (x_{0})⟩_{x_{0}} = 0, for all v ∈ T_{x_{0}}M, all x_{0} ∈ M,  (1.1.14)

which, again because they are Gaussian vectors, implies that ∇_{v}^{γ}W is independent of W (x_{0}) for all v ∈ T_{x_{0}}M, x_{0} ∈ M. (The fact that the processes W (x_{0}) and ∇_{v}^{γ}W (x_{0}) both take values in fibres E_{x_{0}} of a bundle causes no difficulty in using the standard results we used: to reduce to the standard situation where the process takes values in a fixed Hilbert space H_{0}, say, either observe that we can find a measurable trivialization θ : E → M × H_{0} some H_{0}, ⟨⟩_{0} with each θ_{x} : E_{x} → W_{0} an isometry or simply note that we have given to us:

Y : E → M × H

which isometrically maps each E_{x} onto a subspace of H: so we can take H_{0} = H. In this second way we can simply treat Γ(E) as a subspace of the space of maps of M into H.)

Finally, to show that all such metric connections arise this way, let {⟨, ⟩_{x} : x ∈ M} be a smooth metric on E, with a metric connection ˜∇. By Theorem 1.1.2 there is a Euclidean space ℝ^{m}, ⟨⟩ and an X : M ×ℝ^{m} → E whose L-W connection is ˜∇. Let γ_{m} be the standard Gaussian measure of ℝ^{m}, ⟨, ⟩, and let γ be the image measure on Γ(E) of γ under the map ℝ^{m} → Γ(E), e ↦ X(·)e. We claim ∇^{γ} = ˜∇. Indeed if k is the reproducing kernel for γ then if u ∈ E_{x}

k(x, y)u = ∫_{ℝ^{m}}⟨X(x)e, u⟩_{x}X(y)(e) dγ_{m}(e)
= X(y)Y (x)u

for Y (x) = X(x)^{∗}. Thus by (1.1.12) the definitions of the vector fields Z^{u} defined via X and via γ agree and so ∇^{γ} = ˜∇ by their defining property.

Remark 1C. The proof above shows the essential equivalence between the “metric” and “Gaussian” forms. It also shows that the connection depends only on the law, γ, of the process (or equivalently on the subspace H of Γ(E) together with its inner product) not on the process itself. The case of H a Hilbert space of sections is more intrinsic than that of a mapping of a Hilbert space into the space of sections, and often the Gaussian formulation is simpler to use, especially when H is infinite dimensional. However it is often the “metric” form which arises in practice, for example in the gradient systems of example 1B.

Example 1C: Gaussian vector fields on ℝ^{n} are said to be isotropic if they are invariant in law under Euclidean transformations. The covariance of an isotropic Gaussian vector field on ℝ^{n} is determined by two spectral measures F_{L} and F_{N} on ℝ^{+}. It is given by the formula

E (W^{i}(x)W^{j}(y)) = C^{ij}(x − y)  (1.1.15)

with

C^{ij}(z) = ∫_{ℝ^{+}} ∫_{S^{n−1}} e^{iρ⟨z,u⟩} [u_{i}u_{j}σ(du)F_{L}(dρ) + (δ_{ij} − u_{i}u_{j})σ(du)F_{N}(dρ)] ,

σ being the uniform distribution on S^{n−1}. The vector field W (x, ω) has an almost sure C^{∞} version when the measures F_{L} and F_{N} have moments of all orders. From formula (10) in [ELJL97a] or remarks in §1.3 B below (or rather from its generalization to the Gaussian field case) we see that

Γ̆_{j,k}^{i}(x_{0}) = 0.

The connection so constructed is therefore trivial. The isotropic stochastic flows, associated with W was studied in [Jan85] and [BH86]. A special case had first been introduced in [Har81], using an approximation by discrete vortices.

However, even in that isotropic case, nontrivial connections arise if you consider the motion of several points:

In general, the Gaussian field W can be extended into a Gaussian field W^{(d)} on M^{(d)} = {(x_{1}, x_{2}, …, x_{d}) ∈ M^{d}|x_{i} ≠ x_{j}, for i ≠ j} as follows. There is a canonical isomorphism between T_{(x_{1},x_{2},...x_{d})}M^{(d)} and the direct sum ⊕_{α=1}^{d}T_{x_{α}}M. With this identification, we can set W^{(d)}(x_{1}, x_{2}, …, x_{d}) = ⊕_{α=1}^{d}W (x_{α}).

For the isotropic fields on ℝ^{n},

W^{(d)}(x_{1}, x_{2}, …, x_{d}) = ⊕_{α=1}^{d} ∑_{i=1}^{n} W^{i}(x_{α})\frac{∂}{∂x_{i}^{α}} .

It is always a nondegenerate Gaussian vector except in dimension one, when the spectral measure is atomic. See e.g. Darling [Dar92]. We can define an associated metric on (ℝ^{n})^{(d)}

⟨u, v⟩_{(x_{1},...,x_{d})} = ⟨K^{−1}u, v⟩,

where ⟨, ⟩ denotes the Euclidean metric and K^{−1} is the inverse of the matrix of covariances:

K_{(α,i),(β,j)} = C^{ij}(x_{α} − x_{β}), 1 ≤ α, β ≤ n, 1 ≤ i, j ≤ d.

The L-W connection ˘∇^{(d)} is a metric connection on (ℝ^{n})^{(d)} for the metric constructed above. The computations below show it is not the Levi-Civita connection. From the result of §3.3 we can see it is related to a solution to a filtering problem: Given the paths of d points, the restriction to the path of one of the points of the damped adjoint parallel transport along the path of the d-point motion is the restriction of the derivative flow of d-point motion conditioned on the d-point motion.

The connection can be computed explicitly. For example when d = 1, by (1.1.4), the Christoffel symbols of ˘∇^{(d)} are

Γ_{α,β}^{γ} = \frac{∂}{∂t}|_{t=0} 𝔼\{W (x_{γ}) |W (x_{λ} + tδ_{λα}) = δ_{λβ} for all λ\}.

For α ≠ γ, Γ_{α,β}^{γ} clearly vanishes. Moreover

Γ_{α,β}^{α} = \frac{∂}{∂t}|_{t=0} ([K(t, α)]^{−1})_{β,λ} C(x_{α} − x_{λ} + t δ_{λ}^{α})

with ([K(t, α)])_{λ,λ′} = C (x_{λ} − x_{λ′} + tδ_{λ}^{α} − tδ_{λ′}^{α}). Hence, setting K(0)_{λλ′} = C(x_{λ} − x_{λ′})

and K′(0, α) = \frac{∂}{∂t}K(t, α)|_{t=0}

Γ_{α,β}^{α} = (K(0)^{−1}K′(0, α))_{β,α} .

For d = 2 one checks that in particular,

Γ_{12}^{1} = \frac{−C′(x_{1} − x_{2})}{1 − C^{2}(x_{1} − x_{2})}

and

Γ_{11}^{1} = \frac{−C(x_{1} − x_{2}) · C′(x_{1} − x_{2})}{1 − C^{2}(x_{1} − x_{2})}.

1.2 Basic Classes of Examples

Example A: Sums of squares of vector fields: operators in Hörmander form. For a particular manifestation of our basic class of examples consider a second order differential semi-elliptic operator A on M given in the Hörmander form

A = \frac{1}{2} ∑_{1}^{m} L_{X^{j}}L_{X^{j}} + L_{A}.  (1.2.1)

where X^{1}, …, X^{m}, A are smooth vector fields on M, with L_{V} denoting Lie differentiation in the direction of a vector field V. We obtain

X : \underline{ℝ}^{m} → T M

by

X(x)e = ∑_{j}⟨e, e_{j}⟩X^{j}(x)

for e_{1}, …, e_{m} the standard orthonormal base for ℝ^{m}. If we assume E_{x} := X(x)[ℝ^{m}] has constant rank we are in the metric form situation, obtaining a connection ˘∇ on the resulting subbundle E = Image X of T M.

The geometry of such operators have been examined from a different viewpoint (see e.g. [Str86]), usually with a hypoellipticity assumption. They are usually considered as operators on functions. However the presentation of A in the form (1.2.1) shows how to extend it to more general tensors, in particular to differential forms on M. We show below in section 2.4 the relevance of ˘∇ to the analysis of these operators.

Example B: Connections arising in the theory of stochastic flows. Let D be the space of C^{∞} diffeomorphisms of M with C^{∞} topology making it a Polish topological space, (see [Bax84]). Following Baxendale [Bax84] consider a Brownian motion, {ξ_{t} : t ≥ 0}, on D, i.e. a stochastic process on D satisfying

1. almost sure continuity in t,

2. independent increments on the left, i.e. ξ_{t}ξ_{s}^{−1} and ξ_{v}ξ_{u}^{−1} are independent if 0 ≤ s < t ≤ u < v,

3. time homogeneity;

4. ξ_{0}=identity.

For each x ∈ M there is the process {ξ_{t}(x) : t ≥ 0} on M which was shown to be a diffusion with generator A, say. Similarly on M × M there is a process {(ξ_{t}(x), ξ_{t}(y)) : t ≥ 0} for each (x, y) ∈ M × M giving a diffusion with generator A^{2} say. It turns out that for f, g : M → ℝ both C^{∞} with compact support then on f ⊗ g : M × M → ℝ, defined by (x, y) ↦ f(x)g(y),

A^{2} (f ⊗ g) (x, y) = A(f)(x)g(y) + f(x)A(g)(y) + \frac{1}{2}Γ^{ξ} ((df)_{x}, (dg)_{y})

where Γ^{ξ} : T^{∗}M × T^{∗}M → ℝ is the symmetric bilinear map given by

Γ^{ξ} ((df)_{x}, (dg)_{y}) = lim_{h↓0} 𝔼\frac{(f(ξ_{h}(x)) − f(x)) (g(ξ_{h}(y)) − g(y))}{h}.  (1.2.2)

If we assume that A is elliptic, its symbol, which is quadratic and given by Γ^{ξ} ((df)_{x}, (df)_{x}), will be non-degenerate and so determines a metric on T M. Raising and lowering indices of Γ^{ξ}using this metric gives a section k of the bundle L(T M; T M) over M × M,

⟨k(x, y)v_{1}, v_{2}⟩_{y} = Γ^{ξ} (⟨v_{1}, −⟩_{x}, ⟨v_{2}, −⟩_{y}) , v_{1} ∈ T_{x}M, v_{2} ∈ T_{y}M.  (1.2.3)

Following LeJan and Watanabe [LJW84] we can define a connection ∇^{ξ} on T M by

∇_{v}^{ξ}Z = \frac{d}{dt}k(σ(t), x_{0})Z(σ(t)) |_{t=0}  (1.2.4)

just as in equation (1.1.10). Indeed it was shown in [Bax81] (without assuming ellipticity) that Γ^{ξ} is the covariance for a Gaussian measure γ on Γ(T M), mean γ̄, and that there is a correspondence between Brownian motions on D and such Gaussian measures, at least for M compact, see also [Kun90]. Given sufficient regularity the correspondence is obtained via the stochastic differential equation

dx_{t} = ρ_{x_{t}} ∘ dW_{t} + γ̄(x_{t})dt

where {W_{t} : t ≥ 0} is the Wiener process on Γ(T M) associated to γ, and ρ_{x} the evaluation map at x ∈ M, or equivalently for the stochastic differential equation on D for the flow {ξ_{t} : t ≥ 0}

dξ_{t} = (T R_{ξ_{t}}) ∘ dW_{t} + T R_{ξ_{t}}(γ̄)dt

where R_{h} refers to right translation (i.e. composition) by the diffeomorphism h. It was shown in [LJW84] that the generator A is given by

A(f)(x) = \frac{1}{2}tr∇^{ξ}(df)(x) + df(γ̄(x)).

In particular ξ is determined by Γ^{ξ} and γ̄. Given our non-degeneracy assumptions we see ∇^{ξ} = ∇^{γ}, (of course ellipticity can be replaced by constancy of the rank of the symbol).

Remark:

Recall that γ̄ vanishes if and only if ξ_{t} is equal in law to ξ_{t}^{−1} for any fixed t.

Example C: Stochastic differential equations (s.d.e.). The form in which we will most frequently be using the theory will come from stochastic differential equations

dx_{t} = X(x_{t}) ∘ dB_{t} + A(x_{t})dt  (1.2.5)

where {B_{t}} is a R^{m}-valued Brownian motion, A is a smooth vector field and ∘ indicates the integral involved being Stratonovich. For x_{0} in M let ξ_{t}(x_{0}) : 0 ≤ t < ρ(x_{0}) be a maximal solution from x_{0}, so that ρ(x_{0}) ∈ (0, ∞] is the explosion time from x_{0}. There is the associated semigroup P_{t} : t ≥ 0: for bounded measurable functions P_{t}f(x) = 𝔼f(ξ_{t}(x))χ_{t<ρ(x)} with generator A = \frac{1}{2} ∑_{1}^{m} L_{X^{i}}L_{X^{i}} + L_{A}, e.g. Example A above. If M is compact the solutions can be chosen (they are only defined up to sets of measure zero) to give a solution flow of diffeomorphisms as described above: though by taking B_{·} to be a finite dimensional Brownian motion we lose some generality.

The equation (1.2.5) will be said to be regular (or non-singular) if X has constant rank.

Example B: Homogeneous spaces. Suppose we have an action of a Lie group K on M. Let \underline{𝔨} be the Lie algebra of K (taken to be the tangent space at the identity) and exp : \underline{𝔨} → K the exponential map. We have the induced map

X : M × \underline{𝔨} → T M

given by

X(x)(e) = \frac{d}{dt} (exp te · x) |_{t=0}

so that e ↦ X^{e} := X(·)e is a Lie algebra homomorphism. For g ∈ K let L_{g} denote left multiplication by g acting on K or on M and let R_{g} be right multiplication by g acting on K. There is the adjoint action

ad ≡ ad_{K} : K → GL(\underline{𝔨})

given by

ad(g)e = (T R_{g})^{−1} T L_{g}(e), e ∈ \underline{𝔨}, g ∈ K.

Note that if x ∈ M, g ∈ K and e ∈ \underline{𝔨} then

X(gx)e = \frac{d}{dt} exp te · gx |_{t=0} = T L_{g} \frac{d}{dt} g^{−1} (exp te) g · x |_{t=0}  (1.2.6)
= T L_{g} \frac{d}{dt} exp t ad (g^{−1})(e) · x |_{t=0}  (1.2.7)
= T L_{g}X(x) (ad(g^{−1})e) .  (1.2.8)

Suppose further that the action is transitive, and also fixing x_{0} ∈ M the mapping K → M, g ↦ g · x_{0} identifies M with K/H for H = {g ∈ K : g · x_{0} = x_{0}} making M a homogeneous space which is also reductive i.e. there is a linear splitting

\underline{𝔨} = \underline{𝔥} + \underline{𝔪},  (1.2.9)

with \underline{𝔥}∩\underline{𝔪} = 0, where \underline{𝔥} is the Lie algebra of H and \underline{𝔪} some linear subspace of \underline{𝔨} which is ad_{H} invariant:

ad(g) [\underline{𝔪}] = \underline{𝔪}, all g ∈ H.

Note that \underline{𝔥} = ker X(x_{0}), so by (1.2.8), for g ∈ K,

ker X(gx_{0}) = ad (g)[\underline{𝔥}].  (1.2.10)

The reductive property allows us to define \underline{𝔪}_{x} := ad (g) [\underline{𝔪}] ⊂ \underline{𝔨} for x = gx_{0} since if x = ghx_{0}, so h ∈ H, we have

ad(gh) [\underline{𝔪}] = ad (g)ad (h) [\underline{𝔪}] = ad (g)[\underline{𝔪}].

Thus we have the splitting

\underline{𝔨} = KerX(x) + \underline{𝔪}_{x}

for each x ∈ M and we can define Y (x) : T_{x}M → \underline{𝔨} to be the inverse of the restriction of X(x) to \underline{𝔪}_{x}.

Proposition 1.2.1 The connection ˘∇ induced on T M by X, Y is K invariant. If \underline{𝔨} admits an ad_{K}-invariant inner product ⟨, ⟩ for which m is orthogonal to \underline{𝔥} then ˘∇ is the L-W connection for X, ⟨, ⟩.

Remark: This applies to spheres and Grassmanian manifolds in which cases the connections are the Levi-Civita connections. See Theorem 1.4.8 below.

Proof. First observe from (1.2.8) that

ad (g)Y (x)T L_{g^{−1}} : T_{gx}M → \underline{𝔨}

has image in \underline{𝔪}_{gx} and is a right inverse to X(gx). Thus

Y (gx) = ad (g)Y (x)T L_{g^{−1}}.  (1.2.11)

From this and (1.2.8), for g ∈ K, Z a vector field on M, and v ∈ T_{x}M we have

˘∇_{g_{∗}(v)}g_{∗}(Z) = X(gx)d (Y (·)T L_{g}Z (g^{−1}·)) T L_{g}(v)
= T L_{g}X(x) ad (g^{−1})d (Y (·)T L_{g}Z (g^{−1}·)) T L_{g}(v)
= T L_{g}X(x)d (Y (g^{−1}·)Z (g^{−1}·)) T L_{g}(v)
= T L_{g}X(x)d (Y (·)Z(·)) (v)
= g_{∗} ( ˘∇_{v}Z) .

Thus ˘∇ is K-invariant.

When ⟨, ⟩ is ad_{K}-invariant, if \underline{𝔪} is orthogonal to \underline{𝔥} then \underline{𝔪}_{x} ⊥ ker X(x) for each x and we see Y (x) = X(x)^{∗}, from which it follows that ˘∇ is the L-W connection for X, ⟨⟩.

These connections on homogeneous spaces are discussed further in §1.4 below.

1.3 Adjoint connections, torsion skew symmetry, basic formulae

A. From now on in this section we shall assume that E is a subbundle of the tangent bundle T M of M. Then for any linear connection ˜∇ on E there is an operation which gives a differentiation of arbitrary smooth vector fields V in E-directions: define

˜∇′_{u}V = ˜∇_{V (x_{0})}U + [U, V](x_{0}), u ∈ E_{x_{0}}  (1.3.1)

where U is any section of E with U(x_{0}) = u. In terms of the Lie derivative L_{V}, mappings sections U of E into vector fields,

˜∇′V = ˜∇_{V} − L_{V} ,  (1.3.2)

c.f. the tensor A_{V} defined on p.235 of [KN69a]. It is easy to see that ˜∇′V ∈ ΓHom(E, T M), or see [KN69a], p.235, and also that ˜∇′_{U} is a derivation on Γ(M) over C^{∞}(M) for each U ∈ Γ(E). We shall call such an operation a semi-connection on E and call ˜∇′ the adjoint (semi)connection to ˜∇. Adjoint connections were introduced in the case E = T M by Driver in [Dri92].

When E = T M it is a genuine connection: indeed, by definition, in this case for vector fields U, V we have

[U, V] = ˜∇′_{U}V − ˜∇_{V} U  (1.3.3)

whence

˜∇′_{u}V = ˜∇_{u}V − T̃(u, V (x_{0})).  (1.3.4)

Here T̃ : T M × T M → T M is the torsion tensor of ˜∇, defined by

−T̃(u, v) = [U, V](x_{0}) − ˜∇_{u}V + ˜∇_{v}U,  (1.3.5)

where v = V (x_{0}), c.f. Proposition 2.3 of [KN69a].

When E is a genuine subbundle there is still a skew symmetric

T̃ : E ⊕ E → T M

defined by (1.3.5) with U, V now sections of E, (and so, also vector fields on M). For our semi-connection ˜∇′ we can again define T̃′ : E ⊕ E → T M by equation (1.3.5). It is immediate from (1.3.3) and (1.3.5) that T̃′ = −T̃.

Coming back to the specific case of a connection ˘∇ as in §1.1 note for any v, u ∈ E, ˘∇_{v}Z^{u} = ˘∇_{u}Z^{v} ≡ 0. So

T̆(u, v) = −[Z^{u}, Z^{v}].  (1.3.6)

The canonical example of adjoint connections are given on Lie groups G: the adjoint of the flat right invariant connection on T G is the flat left invariant connection (and conversely). More examples are given in §1.3E below.

The name ’semi-connection’ is justified by:

Proposition 1.3.1 Let E^{⊥} be a complementary subbundle to E in T M, so T M = E ⊕ E^{⊥}. Let ∇^{⊥} be any linear connection on E^{⊥} and let ∇^{1} be the direct sum connection induced on T M: ∇^{1} = ˜∇ ⊕ ∇^{⊥}. Let ∇^{1′} be the adjoint connection to ∇^{1}. Then

˜∇′_{u}V = (∇^{1})′_{u}V, u ∈ E_{x_{0}}, V ∈ Γ(T M).

Proof. This follows from equation (1.3.1) since if U ∈ Γ(E) and V ∈ Γ(T M) then ˜∇_{V} U = ∇_{V}^{1} U.

From this proposition we see immediately that it is possible to define operations \frac{D̃′}{∂t} on smooth vector fields \{v_{t} : t ∈ [0, T]\} along piecewise C^{1} curves σ in M with σ̇(t) ∈ E_{σ(t)} for each t (“horizontal” curves) such that: if V ∈ Γ(T M) and v_{t} = V (σ(t)) then

\frac{D̃′v_{t}}{∂t} = ˜∇′_{σ̇(t)}V.

For example simply take ∇^{⊥} as in the proposition to obtain (∇^{1})′ with corresponding \frac{D^{1′}}{∂t} and then restrict to horizontal curves observing the result is independent of the choice of E^{⊥} and ∇^{⊥}. Alternatively there are partially defined “Christoffel symbols”, see Remarks in the next subsection. Similarly there are parallel translations

˜//′_{t} : T_{σ(0)}M → T_{σ(t)}M,

which are linear isomorphisms, but only defined along horizontal curves and indeed, as usual,

\frac{D̃′v_{t}}{∂t} = ˜//′_{t} \frac{d}{dt} ˜//′_{t}^{−1}v_{t}.

A vector field \{v_{t} : t ∈ [0, T]\} along such a curve is parallel if and only if \frac{D̃′v_{t}}{∂t} = 0. Proposition 1.3.1 will enable us to be confident in applying the usual rules of connections to ˜∇′ in consequence. The following extension of (1.3.3) will be of basic importance.

Lemma 1.3.2 For S > 0, T > 0, let σ : [0, S] × [0, T] → M be C^{1} with σ(s, ·) horizontal for each s ∈ [0, S]. Then

\frac{D̃′}{∂t} \frac{∂σ}{∂s} = \frac{D̃}{∂s} \frac{∂σ}{∂t} .  (1.3.7)

Proof. By Proposition 1.3.1, we can assume E = T M. Then

\frac{D̃}{∂t} \frac{∂σ}{∂s} = \frac{D̃}{∂s} \frac{∂σ}{∂t} + T̃(\frac{∂σ}{∂t} , \frac{∂σ}{∂s}).

But, from (1.3.4),

\frac{D̃′}{∂t} \frac{∂σ}{∂s} = \frac{D̃}{∂t} \frac{∂σ}{∂s} − T̃(\frac{∂σ}{∂t} , \frac{∂σ}{∂s}).

In general ˜//′_{t} does not map fibres of E to fibres of E:

Proposition 1.3.3 For the adjoint semi-connection ˜∇′ the following are equivalent:

1. ˜∇′_{u}V ∈ E whenever V ∈ Γ(E) and u ∈ E.

2. ˜//′_{t} maps E_{σ(0)} → E_{σ(t)} whenever σ is a horizontal path.

3. T̃ maps E ⊕ E to E.

4. E is an integrable foliation and ˜∇ restricts to a connection on each of its leaves.

If so ˜∇′ restricts to a connection on the leaves of E.

Proof. Certainly (4) implies (1). Also (1) implies (3) by (1.3.4) and if (3) holds then [U, Z] ∈ Γ(E) when E, Z ∈ Γ(E) by (1.3.5) and (1) holds by (1.3.4), so (3) implies (4). To take in (2) observe that (2) implies (3) using (1.3.4) and the formula \frac{D′v_{t}}{∂t} = ˜//′_{t} \frac{d}{dt} ˜//′_{t}^{−1}v_{t}, while (4) immediately implies (2).

Remark: If P : T M → T M/E is the projection of T M onto the quotient bundle then by (1.3.5)

P (T̃(u^{1}, u^{2})) = −P ([U^{1}, U^{2}](x))  (1.3.8)

for U^{1}, U^{2} ∈ Γ(E) with U^{1}(x) = u^{1}, U^{2}(x) = u^{2}. Thus P ∘ T̃ is independent of the connection on E. It is a well known invariant of the subbundle, e.g. see Strichartz [Str86].

B. Suppose now we have \underline{H} \underrightarrow{Y_{⟵}} E ⟶ 0 as in §1.1A, but still with E a subbundle of T M. The adjoint ˘∇′ of the associated connection ˘∇ will be denoted by ˆ∇, with parallel translation ˆ// etc. It takes on a particularly simple form using the sections {Z^{u} : u ∈ E} of E defined in (1.1.1):

Lemma 1.3.4 1. For any vector field V and u ∈ E_{x_{0}} we have

ˆ∇_{u}V = L_{Z^{u}}V,

2. Let σ : [0, T] → M be a horizontal curve with σ(0) = x_{0}. Then ˆ//_{t}v_{0} = T_{x_{0}}S_{t}^{σ}(v_{0}) all v_{0} ∈ T_{x_{0}}M, where S_{t}^{σ} : M → M, 0 ≤ t ≤ T is the flow of the time dependent vector field Z^{σ̇(t)} = X(·)Y (σ(t))σ̇(t).

Proof. Part (1) is immediate from (1.3.1) or (1.3.2), the defining property of ˘∇, (that ˘∇Z^{u} vanishes at x_{0}), and the skew symmetry of Lie differentiation of vector fields

L_{Z^{u}}V = [Z^{u}, V] = −L_{V} Z^{u}.

For part (2) set x_{t} = S_{t}^{σ}(x_{0}) and v_{t} = T_{x_{0}}S_{t}^{σ}(v_{0}). Since

\{ \begin{matrix}\frac{d}{dt}S_{t}^{σ}(x_{0}) & = & X(S_{t}^{σ}(x_{0}))Y (σ(t))σ̇(t) \\ S_{0}^{σ}(x_{0}) & = & x_{0} \end{matrix}

we see x_{t} = σ(t). Also by Lemma 1.3.2

\{ \begin{matrix}\frac{D̂}{dt}v_{t} & = ˘∇_{v_{t}}X (Y (σ(t))σ̇(t)) = 0 \\ v_{t}|_{t=0} & = v_{0} \end{matrix} .

by the defining property of ˘∇. Thus {v_{t} : 0 ≤ t ≤ T } is parallel for ˆ∇.

Remarks:

(1). Since Lie differentiation with respect to a fixed vector field obeys the usual derivation rules a corollary of Lemma 1.3.4 (1) is that, when E = T M, for any smooth tensor field A on M

ˆ∇_{u}A = L_{Z^{u}}A.  (1.3.9)

For the general case, for u ∈ E we could define ˆ∇_{u}A, for example, by using Proposition 1.3.1 and then (1.3.9) will still hold; or more directly we could use (1.3.9) as the definition.

(2). By expanding U over the basis, Lemma 1.3.4 (1) reads

ˆ∇_{U}V = [Z^{u}, V] = ∑_{1}^{m}[X^{i}, V]⟨X^{i}, U⟩, U ∈ Γ(E), V ∈ Γ(T M);  (1.3.10)

or in the Gaussian form,

ˆ∇_{U}V = 𝔼[W, V]⟨W, U⟩.  (1.3.11)

By (1.3.3),

˘∇_{V} U = [V, U] + 𝔼[W, V]⟨W, U⟩.  (1.3.12)

(3). For X : ℝ^{m} → E ⊂ T M as described let ˜∇ be the Levi-Civita connection for some Riemannian metric on M (or indeed any torsion free connection on T M). Then for U ∈ Γ(E), V ∈ Γ(T M), V (x_{0}) = v, U(x_{0}) = u,

˘∇_{v}U = X(x_{0})d (Y (·)U(·)) (v)
= ˜∇_{v}U + X(x_{0}) ˜∇_{v}Y (U(x_{0}))
= ˜∇_{v}U + Γ̃(v, u),

say. Also by definition, (1.3.1),

ˆ∇_{u}V = ˘∇_{v}U + [U, V](x_{0})
= ˜∇_{v}U + Γ̃(v, u) + ˜∇_{u}V − ˜∇_{v}U
= ˜∇_{u}V + Γ̃(v, u).

Working in a chart (U, ϕ) of M with X^{ϕ} the local representation of X

X^{ϕ} = T ϕ ∘ X : ϕ(U) × ℝ^{m} → T ϕ[E] ⊂ ϕ(U) × ℝ^{n}

etc. and taking ˜∇ to be the usual differentiation in ℝ^{n} this shows that ˘∇ has “Christoffel symbols” given by

Γ_{x_{0}}^{ϕ}(ṽ, ũ) = X^{ϕ}(x_{0})DY^{ϕ}(x_{0})(ṽ)(ũ)  (1.3.13)

where now ṽ ∈ ℝ^{m} and ũ ∈ T_{x_{0}}ϕ(E_{x_{0}}), x_{0} ∈ U. Moreover

ˆ∇_{u}V = DV^{ϕ} (ϕ(x_{0})) ũ + Γ_{x_{0}}^{ϕ} (T_{x_{0}}ϕ(v), T_{x_{0}}ϕ(u))  (1.3.14)

Equivalently in the nondegenerate case

Γ̆_{jk}^{i} = − ∑_{r=1}^{m} ∑_{l=1}^{n} \frac{∂X(x_{0})^{r,i}}{∂x^{j}} X(x_{0})^{r,ℓ}g_{kℓ},  (1.3.15)

where {X(x)^{r,i}}, {1 ≤ i ≤ n}, {1 ≤ r ≤ m} is the matrix representing X(x) : ℝ^{m} → ℝ, i.e. X(x)^{r,i} = ⟨X(e_{r}), f_{i}⟩ for {e_{i}} and {f_{i}} orthonormal bases for ℝ^{m} and T_{x}M respectively, and {g_{kℓ}} the metric tensor. This shows that ˘∇ is the L-W connection defined in [LJW84].

C. It will be important to know when ˆ∇ is adapted to some Riemannian metric ⟨, ⟩′ on T M, (see also §2.1) in the sense that

d ⟨Z_{1}(·), Z_{2}(·)⟩′ (u) = ⟨ ˆ∇_{u}Z_{1}(x), Z_{2}(x)⟩′_{x} + ⟨Z_{1}(x), ˆ∇_{u}Z_{2}(x)⟩′_{x}  (1.3.16)

for u ∈ E_{x} and all Z_{1}, Z_{2} ∈ Γ(T M), or equivalently that parallel translation ˆ//_{t} along any smooth horizontal curve preserves ⟨, ⟩′. In the case that E = T M and ⟨, ⟩′ = ⟨, ⟩, when this holds ˘∇ is said to be torsion skew symmetric. See §2.2. Let ∇′ be the Levi-Civita connection for ⟨, ⟩′.

Proposition 1.3.5 The adjoint ˆ∇ is adapted to ⟨, ⟩′ if and only if ∇′Z^{u} is skew symmetric on T_{x}M, ⟨, ⟩′_{x} for all u ∈ E_{x}, x ∈ M.

Proof. By Lemma 1.3.4 (2), ˆ∇ is adapted to ⟨, ⟩′ if and only if

\frac{d}{dt} ⟨T S_{t}^{σ}(v_{0}^{1}), T S_{t}^{σ}(v_{0}^{2})⟩′_{σ(t)} |_{t=0} = 0

for all horizontal curves σ and v_{0}^{1}, v_{0}^{2} in T_{σ(0)}M. But this is precisely the condition

⟨∇′_{v_{0}^{1}}Z^{σ̇(0)}, v_{0}^{2}⟩′_{σ(0)} + ⟨v_{0}^{1}, ∇′_{v_{0}^{2}}Z^{σ̇(0)}⟩′_{σ(0)} = 0,

since \frac{D′}{∂t} T S_{t}^{σ}(v_{0}^{1}) = ∇′Z^{σ̇(0)} if D′ is differentiation with respect to ∇′.

Corollary 1.3.6 Suppose X(x) is injective for each x ∈ M. Then ˆ∇ is adapted to some Riemannian metric ⟨, ⟩′ on T M if and only if X^{e} ≡ X(·)(e) is an infinitesimal isometry for each e ∈ H.

Proof. Injectivity implies that X^{e} = Z^{u} for u = X(x_{0})e, any x_{0} ∈ M. But skew-symmetry of ∇′X^{e} is equivalent to X^{e} being an infinitesimal isometry (e.g. see [KN69a], p.237).

Remark 2C. Note that in the Gaussian form, §1.1C, the injectivity hypothesis becomes the assumption that the vector fields in the reproducing kernel Hilbert space H_{γ} of γ never vanish. In this case ˘∇ is the trivial connection determined by the trivialization Y of E and so the curvature R̆ of ˘∇ vanishes (alternatively ˘∇W vanishes and R̆ is seen to vanish by the expression given in Appendix I of [ELJL97a], see also Proposition C.4 in Appendix C). In the stochastic flow picture, §1.2F, it implies that for any x_{0} ∈ M, and T > 0 the infinite dimensional process {ξ_{t} : 0 ≤ t ≤ T } can be expressed in terms of {ξ_{t}(x_{0}) : 0 ≤ t ≤ T }. The standard probabilistic approach to second order elliptic operators on ℝ^{n} is to use such X taken to be the positive square root of the symbol of the operator considered as a map of ℝ^{n} into the positive definite symmetric matrices.

Corollary 1.3.7 Suppose X is injective but the Lie algebra generated by {X^{e} : e ∈ H} has dimension greater than \frac{1}{2}n(n + 1) when n = dimM < ∞. Then ˆ∇ is not adapted to any Riemannian metric on M.

Proof. The Lie algebra of infinitesimal isometries of a connected Riemannian manifold has dimension at most \frac{1}{2}n(n + 1) ([KN69a],Theorem 3.3 on p238).

Example 2C. For M = ℝ^{2} define vector fields X^{1}, X^{2} by X^{1}(x, y) = \frac{∂}{∂x}, X^{2}(x, y) = x^{3} \frac{∂}{∂x} + \frac{∂}{∂y}. This gives an injective X as in §1E and the Lie algebra generated by X^{1}, X^{2} is easily seen to be infinite dimensional. Thus the induced adjoint connection ˆ∇ is not adapted to any metric. This example could easily be modified outside of a compact set to make it periodic, and so project to a compact surface.

D. Consider the stochastic differential equation

dx_{t} = X(x_{t}) ∘ dB_{t} + A(x_{t})dt,  (1.3.17)

where {B_{t}} is a R^{m}-valued Brownian motion, A is a smooth vector field and ∘ indicates the integral involved being Stratonovich. Let {ξ_{t}(x_{0})} be the solution with initial value x_{0} and ˘∇ the connection constructed from X. Then roughly speaking the parallel translation along the paths of {ξ_{t}} is bounded if and only if ˆ∇ is adapted to some Riemannian metric. (This statement is true for smooth paths as easily seen from the proof of the next theorem). See also Theorem 5.0.7 for the corresponding result on the derivative flow.

Theorem 1.3.8 Assume that X is nondegenerate and the stochastic differential equation (1.3.17) does not explode. For x_{0}, y_{0} ∈ M and T > 0, let {ξ_{t}^{T,y_{0}}(x_{0}) : 0 ≤ t ≤ T } be the process conditioned to be y_{0} at time T of {ξ_{t}(x_{0})} and let ˆ//_{t} =: ˆ//_{t}^{T,x_{0},y_{0}} be parallel translation along {ξ_{t}^{T,y_{0}}(x_{0})}. If ˆ∇ is adapted to some Riemannian metric then ˆ//_{T} is a bounded L(T_{x_{0}}M, T_{y_{0}}M) valued random variable for such x_{0}, y_{0} and T > 0. Conversely if just for one set of x_{0}, y_{0} and T > 0 the parallel translation process ˆ//_{T} along the path of {ξ_{t}^{T,y_{0}}(x_{0})} is bounded then ˆ∇ is adapted to some Riemannian metric on M.

Proof. The ‘if’ part is clear since ˆ//_{T} would be an isometry.

Suppose that ˆ//_{T} is bounded. Let u_{0} be a frame at x_{0} and P (u_{0}) the holonomy bundle through u_{0}, i.e.

P (u_{0}) = {u ∈ GL(M) | there exists a horizontal curve from u_{0} to u.},

with structure group

Φ(u_{0}) = {g ∈ GL(n) | u_{0} · g ∈ P (u_{0})}.

We can reduce ˆ∇, now a genuine connection, to a connection on P (u_{0}). Let (u_{t}) be the solution to

du_{t} = X̃(u_{t}) ∘ dB_{t} + Ã(u_{t})dt

with initial value u_{0}. Here X̃ and Ã are the horizontal lifts in P (u_{0}) of X and A. Then {u_{t} : 0 ≤ t ≤ T } is the horizontal lift of {ξ_{t}(x_{0}) : 0 ≤ t ≤ T }.

The support of the law μ_{t}(P) of u_{T} is all of P (u_{0}) by the Stroock-Varadhan support theorem and the definition of P (u_{0}). Consequently by Carverhill [Car88] the support of u_{T} when {ξ_{T} (x_{0})} is conditioned to have ξ_{T} (x_{0}) = y_{0} is π_{0}^{−1}(y_{0}), where π_{0} : P (u_{0}) → M is the projection. Thus parallel translations from T_{x_{0}}M to T_{y_{0}}M along the paths of the conditioned process {ξ_{T}^{y_{0}}(x_{0}) : 0 ≤ t ≤ T } are dense in the space of parallel translations along smooth paths. So the latter is a bounded set and Φ(u_{0}) is bounded in GL(n). As a consequence there is an inner product on ℝ^{n}, ⟨, ⟩′ say, invariant under Φ(u_{0}). The required metric at a point z of M is then ⟨v^{1}, v^{2}⟩_{z} = ⟨u^{−1}v^{1}, u^{−1}v^{2}⟩′ for any u ∈ π_{0}^{−1}(z).

E. Proposition 1.3.3 was concerned with the case when E is integrable. At the other extreme is the situation where the vector fields X^{1}, …, X^{m} together with their iterated brackets span T_{x}M for each x in M, giving hypoellipticity of the operator A by Hörmander’s theorem. Bismut showed how this hypoellipticity was reflected in the behaviour of the derivative of the associated stochastic flow so that Lemma 1.3.4 (2) makes it not surprising that it is also reflected in the behaviour of parallel translation ˆ//_{s} along the paths of the associated diffusion, as we see next. As before we consider the stochastic differential equation (1.3.17) and assume it to be regular with A(x) ∈ E_{x} for each x in M. Our discussion is an adaptation of that in [Bel87] which was in turn based on [Bis81].

Let R_{t}(ω) = span{ ˆ//_{s}^{−1}X(x_{s})e : e ∈ ℝ^{m}, 0 ≤ s ≤ t ∧ τ} ∈ T_{x_{0}}M where τ is a fixed, positive predictable stopping time less than the explosion time for our stochastic differential equation. By Proposition 1.3.3 in the integrable case R_{t}(ω) = E_{x_{0}} for each t ≥ 0. In general let \overline{E}_{x_{0}} ⊂ T_{x_{0}}M be the linear span of X^{1}(x_{0}), …, X^{m}(x_{0}) together with all the brackets and iterated brackets of the vector fields X^{1}, …, X^{m} evaluated at x_{0} (this depends only on E ↪ T M, not on ˘∇ or a choice of X^{1}, …, X^{m} determining E):

Theorem 1.3.9 For each t > 0,

\overline{E}_{x_{0}} ⊂ R_{t}(ω) almost all ω in Ω.

Proof. Set R(ω) = ∩_{t>0}R_{t}(ω). By the Blumenthal 0-1 law there exists a nonrandom R_{0} ⊂ T_{x_{0}}M with R_{0} = R(ω) almost surely. Moreover there therefore exists a predictable stopping time τ_{1} with 0 < τ_{1} < τ such that R_{t}(ω) = R_{0} for 0 ≤ t ≤ τ_{1} almost surely.

Suppose ℓ ∈ T_{x_{0}}^{∗}M annihilates R_{0}. Then for e ∈ ℝ^{m}, with X^{e} := X(·)e,

ℓ( ˆ//_{s}^{−1}X^{e}(x_{s})) = 0, 0 ≤ s ≤ τ_{1}.

Now if Z is a vector field with ℓ( ˆ//_{s}^{−1}Z(x_{s})) = 0, 0 ≤ s ≤ τ_{1}, taking the local martingale part of its canonical decomposition we have

ℓ( ˆ//_{s}^{−1} ˆ∇_{X(x_{s})f}Z) = 0

all f ∈ ℝ^{m}, 0 ≤ s ≤ τ_{1}.

By Lemma 1.3.4 (1) this gives

ℓ( ˆ//_{s}^{−1}[Z, X^{f}](x_{s})) = 0,  0 ≤ s ≤ τ_{1},  (1.3.18)

since [Z, X^{f}] = [Z, Z^{X^{f}}] + ˘∇_{Z}X^{f} and ˆ//_{s}^{−1} ˘∇_{Z}X^{f} ∈ R_{0} because ˘∇_{Z}X^{f} ∈ E_{x_{s}}. Taking Z = X^{e} show that [X^{e}, X^{f}](x_{0}) ∈ R_{0} each e, f ∈ ℝ^{m}. Repeating the argument, using (1.3.18) with Z = [X^{e}, X^{f}] gives [ [X^{e}, X^{f}], X^{g}](x_{0}) ∈ R_{0} for all e, f, g in ℝ^{m}, and the full result follows by induction.

1.4 Example: Homogeneous spaces continued

A. For an important class of examples we will go back to situation of a reductive homogeneous space M = K/H described in §1.2 using the notation there. In Proposition 1.2.1 we saw from our construction that the action of K determines a K-invariant connection on M and that given an ad_{K}-invariant inner product on \underline{𝔨} the L-W connection is K-invariant: in particular the metric ⟨ ⟩^{X} (i.e. induced by X(x) : \underline{𝔨} → T M ) is K-invariant, as is seen from (1.2.8). In general given an inner product ⟨ ⟩ on \underline{𝔨} the metric ⟨ ⟩^{X} induced on T M will not be K-invariant.

Proposition 1.4.1 A K-invariant connection ˜∇ on M is uniquely determined by the mapping

\underline{𝔪} → L(T_{x_{0}}M; T_{x_{0}}M)

given by

e ↦ ˜∇′X^{e}|_{T_{x_{0}}M} ,

where ˜∇′ is the adjoint connection.

Proof. Observe that for any vector field Z on M

Ã_{Z} : T M → T M

defined by Ã_{Z}(v) := − ˜∇_{v}Z − T̃(Z, v) as defined in [[KN69a], p255, [KN69b], p188] is given by Ã_{Z}(v) = ˜∇′_{v}Z. The result is then a reformulation of Corollary 2.2 of [[KN69b], p191].

B. The K-invariant connection on M corresponding to the identically zero map: \underline{𝔪} → L(T_{x_{0}}M; T_{x_{0}}M) is the canonical connection

Theorem 1.4.2 Let ˘∇ be the connection on M determined as in §1.2 by its reductive homogeneous space structure and let ˆ∇ be its adjoint. Then ˆ∇ is the canonical connection. In particular

(i). R̂(u, v)w = − [[Z^{u}, Z^{v}] − Z^{[Z^{u},Z^{v}]}, Z^{w}] , u, v, w ∈ T_{x_{0}}M,

(ii). ˆ∇T̂ = 0,

(iii). ˆ∇R̂ = 0.

Proof. Take ˜∇ = ˆ∇ in Proposition 1.4.1 and use the defining property of ˘∇ to see ˜∇ is the canonical connection. For (i), (ii) and (iii) see Theorem 2.6 p193 [KN69b] and use the fact that e ↦ X^{e}(·) is a homomorphism of Lie algebras.

Corollary 1.4.3 Any K-invariant tensor on M is ˆ∇-parallel.

Proof. See Proposition 2.7 p192 of [KN69b].

From this we immediately have

Corollary 1.4.4 The connection ˆ∇ is metric for any K-invariant metric on M.

Taking ˜∇ = ˆ∇ we can obtain in this way a class of metric connections whose adjoints are not metric for any metric on M.

C. Suppose next that as well as being reductive there is an ad_{H}-invariant inner product B on \underline{𝔪}. We then obtain a K-invariant Riemannian metric on M which agrees with B under the isomorphism \underline{𝔪} → T_{x_{0}}M. The space, together with B, is called naturally reductive if it has a decomposition as before with also

B (α, [β, γ]\underline{𝔪}) + B ([β, α]\underline{𝔪} , γ) = 0, α, β, γ ∈ \underline{𝔪}  (1.4.1)

where the subscript \underline{𝔪} refers to the projection in \underline{𝔨} onto \underline{𝔪}.

Proposition 1.4.5 The decomposition \underline{𝔨} = \underline{𝔪} + \underline{𝔥} together with B is naturally reductive if and only if ˆ∇ is torsion skew symmetric for the induced K-invariant metric on M.

Proof.

⟨T̂(u, v), w⟩_{x_{0}} = − ⟨[Z^{u}, Z^{v}] (x_{0}), w⟩_{x_{0}} , u, v, w, ∈ T_{x_{0}}M

by (1.3.6). Set α = Y (x_{0})u, β = Y (x_{0})v, γ = Y (x_{0})w. Then

[Z^{u}, Z^{v}] (x_{0}) = [X^{α}, X^{β}] (x_{0})
= X(x_{0}) ([α, β])
= X(x_{0}) ([α, β]\underline{𝔪})

and the result follows from (1.4.1).

Corollary 1.4.6 In the naturally reductive case ˘∇ is a torsion skew symmetric connection.

Note however that the metric involved may not be induced, via X, by an inner product on \underline{𝔨}. However from [KN69b] p203 Theorem 3.5 we see that if \underline{𝔨} has an ad_{K}- invariant inner product ⟨ ⟩, we can let \underline{𝔪} = ker X(x_{0})^{⊥} = \underline{𝔥}^{⊥} to have K/H naturally reductive with B(α, β) = ⟨α, β⟩, α, β ∈ \underline{𝔪}. Thus

Theorem 1.4.7 Let \underline{𝔨} have an ad_{K}-invariant inner product ⟨ ⟩. Then the L-W connection on M determined by X, ⟨⟩ is torsion skew symmetric and K-invariant. Its adjoint connection is the canonical connection of the corresponding reductive homogeneous space structure.

D. Specializing further suppose that we have a symmetric space (K, H, σ): so K, H are as before with a reductive decomposition \underline{𝔨} = \underline{𝔪} + \underline{𝔥} such that \underline{𝔪} and \underline{𝔥} are, respectively, the −1 and +1 eigenspaces for the involution σ of \underline{𝔨}.

Theorem 1.4.8 For a symmetric space ˘∇ and ˆ∇ are torsion free. In particular they are the Levi-Civita connection of any K-invariant Riemannian metric on M.

Proof. That T̆ = −T̂ = 0 follows from (1.3.6) and the fact that [\underline{𝔪}, \underline{𝔪}] ⊂\underline{𝔥}, see Proposition 2.1 p226 [KN69b], or from the corresponding fact for the canonical connection [KN69b] p231 Theorem 3.2. The result follows from Corollary 1.4.4.

Corollary 1.4.9 Let the symmetric space (K, H, σ) be such that \underline{𝔨} has an ad_{K}-invariant inner product ⟨ ⟩ invariant under σ. Then the L-W connection for X, ⟨⟩ is the Levi-Civita connection for the induced metric (which is K-invariant).

Proof. In this situation \underline{𝔪} ⊥ \underline{𝔨}, see [[KN69b], p233], and so \underline{𝔪}_{x} ⊥ KerX(x) for each x ∈ M: thus Y = X^{∗} and ˘∇ is the L-W connection and hence metric. Since it is torsion free it is Levi-Civita.

Example 1.4.1 Lie groups as symmetric spaces

Recall the standard symmetric space structure for a Lie group G. Let ΔG = {(g, g) ∈ G × G : g ∈ G}. Let G × G act on G by

(g, h) · x = gxh^{−1}.

The stabilizer of e is ΔG and G has the homogeneous space structure G = G×G/ΔG, and symmetric space structure with involution induced from σ : G × G → G × G given by σ(g, h) = (h, g). For example the symmetry s_{e} : G → G is just x ↦ x^{−1}, see [KN69b] (p228).

The relevant decomposition of \underline{𝔨} = 𝔤 ⊕ 𝔤 is

\underline{𝔨} = Δ𝔤 ⊕ \underline{𝔪}

where Δ𝔤 = \{(α, α) : α ∈ 𝔤\}, \underline{𝔪} = \{(α, −α) : α ∈ 𝔤\}, see [KN69b] (p198). Now suppose 𝔤 and hence 𝔤 × 𝔤 has an ad_{G}-invariant inner product ⟨ ⟩.

The stochastic differential equation on G is

dx_{t} = T L_{x_{t}} ∘ dB_{t} − T R_{x_{t}} ∘ dB′_{t}

for (B_{·}), (B′_{·}) independent BM(𝔤). The flow is given by ξ_{t}(x) = g_{t}xg_{t}^{′−1} for

dg_{t} = T R_{g_{t}} ∘ dB_{t}
dg′_{t} = T R_{g_{t}} ∘ dB′_{t}

with g_{0} = e = g′_{0}.

The flow consists of isometries so that the moment exponents are zero. However for G with bi-invariant metric (as we are considering), if 𝔤 has trivial centre then Ric > 0. Thus we obtain a class of stochastic differential equations such that

(i) Ric̆ > 0; (ii) all moment exponents vanish; (iii) ξ_{·} consists of isometries; (iv) ˘∇ is the Levi-Civita connection. In particular (i) and (ii) can be contrasted with results which says that negative curvature implies first moment exponent positive. They also give an example where the hypotheses of Corollary 6.4.7 in [Li92] hold.

Remark: Among other homogeneous spaces with a Riemannian space structure are the spheres S^{n} = SO(n + 1)/SO(n), oriented Grassmannian manifolds SO(p + q)/SO(p) × SO(q) and hyperbolic spaces O(1, n)/SO(n).

The infinitesimal generators and associated operators

For any second order elliptic operator L with smooth coefficients and L1 ≡ 0 on a manifold M of dimension greater than 1, we construct X : \underline{ℝ}^{m} → T M, for some m, such that L = \frac{1}{2} ∑_{1}^{m} L_{X^{j}}L_{X^{j}}, the differential generator for the solutions of the stochastic differential equation without drift dx_{t} = X(x_{t}) ∘ dB_{t}. This result is, in fact, proved for a class of semi-elliptic operators. In section 2.4 the Hölmander form operator A^{q} = \frac{1}{2} ∑_{j} L_{X^{j}}L_{X^{j}} + L_{A} on differential q-forms is analysed. Section §2.2 discusses a special class of connections on T M and also the natural generalization δ̂ of δ to our situation. We show in §2.4 that A^{q} = −\frac{1}{2}(dδ̂ + δ̂d) + L_{A} so that when the L-W connection associated to X is the Levi-Civita connection and A = 0 then A^{q} is the De Rham-Hodge Laplacian. In the regular case there is the Weitzenbock formula:

A^{q}ϕ = \frac{1}{2}tr_{E} ˆ∇_{·}( ˆ∇_{·}ϕ) + L_{A}ϕ − \frac{1}{2}R̆^{q}ϕ

for R̆^{q} a zero order term, the ’Weitzenbock curvature’ related to ˘∇. In the last section we give conditions for the leading order terms of A^{q} to be symmetrizable as an operator on a suitable L^{2} space. In particular we show that this holds if A^{0} is symmetrizable and ˆ∇ is metric for some metric on T M.

2.1 The irrelevance of drift in dimension greater than 1

A. The first application of the construction in section §1.1 is that any elliptic differential operator can be considered as an infinitesimal generator to some stochastic differential equation with zero drift: dx_{t} = X(x_{t}) ∘ dB_{t}. Let E be a subbundle of T M with fibre dimension p. Let Z be a E-valued vector field, and V : ℝ^{m} → E ⊂ T M a C^{∞} surjection. Consider the operator L on M:

L = \frac{1}{2} ∑_{1}^{m} L_{V^{i}}L_{V^{i}} + L_{Z},  (2.1.1)

where V^{i} = V (e^{i}) for an orthonormal base of ℝ^{m}.

Theorem 2.1.1 Assume p > 1. For L as given above, there is a map X : M ×ℝ^{m} → T M linear in the second variable such that the solution to dx_{t} = X(x_{t}) ∘ dB_{t} has L as infinitesimal generator, i.e. L = \frac{1}{2} ∑_{j} L_{X^{j}}L_{X^{j}}.

Given E the Riemannian metric induced by V. Recall that a diffusion {ξ_{t}(x) : 0 ≤ t < ρ(x)} generator L is a ˘∇ martingale if f(ξ_{t}(x)) − trace_{E} ˘∇df(ξ_{t}(x)) is a local martingale for any smooth f : M → ℝ. (e.g. [Eme89]). An immediate corollary of the Theorem is that a diffusion {ξ_{t}(x) : 0 ≤ t < ρ(x), x ∈ M}, whose generator satisfies the conditions of Theorem 2.1.1 is a ˘∇-martingale for some metric connection ˘∇ on E. In fact (as will be seen explicitly in §3.3), ξ_{t}(x_{0}) will be the stochastic development of a Brownian motion on E_{x_{0}} using ˘∇. The existence of a connection on T M for which this is true for a non-degenerate diffusion was shown by Ikeda-Watanabe [IW89] and a modification of their construction for the case E = T M is one of the main ingredients of our proof. The other key ingredient is Narasimhan & Ramanan’s theorem on universal connections used via Theorem 1.1.2.

The proof is given in §§C below. To adapt Ikeda & Watanabe’s construction we need the modification of the classical result on connections given in Proposition 2.1.2 below. Its proof taken up §§B. Some of the notation in §§B will be used later.

B. Let E be a subbundle of T M with a given metric. Two connections ∇^{a} and ∇^{b} on E are associated with a bilinear map D^{ab} : T M × E → E such that

∇_{V}^{a} U = ∇_{V}^{b} U + D^{ab}(V, U), V ∈ T M, U ∈ E.

Let T^{a} and T^{b} be respectively the torsions, defined by (1.3.5), for the two connections, then

T^{a}(u, v)) = T^{b}(u, v) + D^{ab}(u, v) − D^{ab}(v, u), u, v ∈ E.  (2.1.2)

In this section we use uppercase letters for vector fields and lowercase letters for tangent vectors.

It will be convenient to have a class of connections on E with which to relate metric connections. In the non-degenerate case the obvious base connection is the Levi-Civita connection ∇ for the given metric. In the degenerate, regular, case let ⟨, ⟩^{0} be an extension to T M of the metric ⟨, ⟩ on E and let P_{E} : T M → E be the corresponding orthogonal projection. Let ∇^{0} be the connection on E which is the push forward by P_{E} of the Levi-Civita connection ∇ of ⟨, ⟩:

∇_{V}^{0} U = P_{E}∇_{V} U,  U ∈ Γ(E), V ∈ Γ(T M).  (2.1.3)

In fact ∇^{0} is a metric connection. Note that if T^{0} : T × E → T M is the torsion for ∇^{0}, defined by (1.3.5) or rather its modification for connections on the subbundle E,

T^{0}(U, V) = P_{E}(∇_{U}V − ∇_{V} U) − [U, V] = −(I − P_{E})[U, V].  (2.1.4)

Let T̃ be the torsion for a connection ˜∇ on E and take (I − P_{E}) of (1.3.5) to see:

(I − P_{E})T̃(U, V) = −(I − P_{E})[U, V] = T^{0}(U, V),  (2.1.5)

and so

T̃(U, V) = P_{E}T̃(U, V) − (I − P_{E})[U, V].  (2.1.6)

The converse also holds:

Proposition 2.1.2 Let T̃ : E×E → T M be a skew symmetric map satisfying (2.1.5). There is a metric connection ˜∇ on E with T̃ as its torsion.

The proof of Proposition 2.1.2 will be given after Lemma 2.1.3. First we introduce the tensors D̃ and S. Define D̃ : T M × E → E by

˜∇_{V} U = ∇_{V}^{0} U + D̃(V, U).  (2.1.7)

For U, V ∈ ΓE, write D̃ as the sum of its symmetric part S and antisymmetric part A:

D̃(u, v) = A(u, v) + S(u, v).

By (2.1.2), T̃(u, v) = T^{0}(u, v) + 2A(u, v). So

A(u, v) = \frac{1}{2}P_{E}T̃(u, v),

so that

˜∇_{v}U = ∇_{v}^{0}U + \frac{1}{2}P_{E}T̃(u, v) + S(u, v), u, v ∈ ΓE.  (2.1.8)

Let Cyl denote cyclic sum.

Lemma 2.1.3 A connection ˜∇ on E is metric if and only if the map D̃(v, ·) : E → E is skew symmetric for each v ∈ T M,

⟨D̃(v, u_{1}), u_{2}⟩ + ⟨D̃(v, u_{2}), u_{1}⟩ = 0,  u_{1}, u_{2} ∈ E.  (2.1.9)

For V ∈ ΓE, (2.1.9) is equivalent to

⟨S(u_{1}, u_{2}), v⟩ = \frac{1}{2} ⟨P_{E}T̃(v, u_{1}), u_{2}⟩ + \frac{1}{2} ⟨P_{E}T̃(v, u_{2}), u_{1}⟩  (2.1.10)

and (2.1.10) implies

Cyl ⟨S(·, ·), ·⟩ = 0.  (2.1.11)

Consequently for U_{1}, U_{2}, and V in ΓE,

\begin{matrix}⟨D̃(V, U_{1}), U_{2}⟩ \\ = \frac{1}{2} \{⟨P_{E}T̃(V, U_{1}), U_{2}⟩ + ⟨P_{E}T̃(U_{2}, V), U_{1}⟩ + ⟨P_{E}T̃(U_{2}, U_{1}), V ⟩\} \end{matrix}  (2.1.12)

Proof. First take V ∈ Γ(T M) and U_{i} ∈ Γ(E). The equivalence of ˜∇ being metric and the skew symmetricity of D̃(−, ·) follows from differentiating ⟨U_{1}, U_{2}⟩:

d_{V} ⟨U_{1}, U_{2}⟩
= ⟨∇_{V}^{0} U_{1}, U_{2}⟩ + ⟨U_{1}, ∇_{V}^{0} U_{2}⟩
= ⟨ ˜∇_{V} U_{1}, U_{2}⟩ + ⟨U_{1}, ˜∇_{V} U_{2}⟩ − ⟨D̃(V, U_{1}), U_{2}⟩ − ⟨U_{1}, D̃(V, U_{2})⟩ .

So ˜∇ is metric if and only if

⟨D̃(V, U_{1}), U_{2}⟩ + ⟨U_{1}, D̃(V, U_{2})⟩ = 0.

Next suppose V and U_{i} ∈ ΓE, writing D̃ = A + S to get:

\begin{matrix}⟨A(V, U_{1}), U_{2}⟩ + ⟨A(V, U_{2}), U_{1}⟩ \\ = − ⟨S(V, U_{1}), U_{2}⟩ − ⟨S(V, U_{2}), U_{1}⟩ \end{matrix}  (2.1.13)

Suppose (2.1.13) holds. Observe for an alternating bilinear map L : E × E → E:

Cyl [⟨L(v, u_{1}), u_{2}⟩ + ⟨L(v, u_{2}), u_{1}⟩] = 0.  (2.1.14)

Take the cyclic sums of equation (2.1.13) and apply (2.1.14) to A to obtain

Cyl ⟨S(V, U_{1}), U_{2}⟩ = 0.

Substitute the above back to (2.1.13) to see:

⟨A(V, U_{1}), U_{2}⟩ + ⟨A(V, U_{2}), U_{1}⟩ = ⟨S(U_{1}, U_{2}), V ⟩ .

So (2.1.13), and therefore (2.1.9), implies (2.1.10). On the other hand (2.1.11) clearly follows from (2.1.10) and the two give (2.1.9).

Proof of Proposition 2.1.2. Define S : E ×E → T M by (2.1.10). Set D̃(v, u) = 0 for v ∈ (ΓE)^{⊥}, and D̃(v, u) = \frac{1}{2}P_{E}T̃(v, u) + S(v, u) for v ∈ ΓE. Define ˜∇ by

˜∇_{V} U = ∇_{V}^{0} U + D̃(V, U).

Then the equality (2.1.9) holds for v ∈ ΓE and extends to ΓT M since by the construction D̃(v, u) = D̃(P_{E}v, u). The connection ˜∇ is the required connection.

C. Proof of Theorem 2.1.1. Let ∇ be the Levi-Civita connection for a metric extending the metric induced on E by the map V and ∇^{0} its projection to E as before. Set Z̄ = \frac{1}{2} ∑_{1}^{m} ∇_{V^{i}}^{0}V^{i} + Z and choose T̃ such that

trace_{E} ⟨P_{E}T̃(u, −), −⟩ = −2 ⟨Z̄(x), u⟩ , u ∈ E_{x}.  (2.1.15)

One choice of such T̃ is given by

P_{E}T̃(v, u) = \frac{2}{p − 1}(u ∧ v)Z̄(x), u, v ∈ E_{x}.

Recall (u ∧ v)Z̄(x) = ⟨Z̄(x), u⟩ v − ⟨Z̄(x), v⟩ u. Let ˜∇ be the metric connection on E with torsion T̃ : E × E → T M as constructed in Proposition 2.1.2. We show the associated stochastic differential equation has L as infinitesimal generator.

Let X : \underline{ℝ}^{m} → E ⊂ T M be a bundle map, as in Theorem 1.1.2, which gives rise to the metric connection ˜∇. The solution to the following stochastic differential equation

dx_{t} = X(x_{t}) ∘ dB_{t}  (2.1.16)

has generator

A^{0} = trace_{E}[∇^{0}]^{2} + \frac{1}{2} ∑_{i=1}^{m} ∇^{0}X^{i}(X^{i}),

On the other hand from (2.1.1)

L = \frac{1}{2}trace_{E}[∇^{0}]^{2} + ( \frac{1}{2} ∑_{1}^{m} ∇_{V^{i}}^{0}V^{i} + Z) = \frac{1}{2}trace_{E}[∇^{0}]^{2} + Z̄.

The required result then follows after we show

∑_{i=1}^{m} ∇^{0}X^{i}(X^{i}) = − ∑_{1}^{m} D̃(X^{i,}X^{i}) = tr_{E}D̃(−, −)

equals Z̄. For this note that for all v ∈ E,

⟨∑ D̃(X^{i}, X^{i}), v⟩ = ⟨∑ S(X^{i}, X^{i}), v⟩

= − ∑_{i} ⟨P_{E}T̃(v, X^{i}), X^{i}⟩ = −2 ⟨Z̄, v⟩ .

Consequently

trace_{E}[∇^{0}]^{2} + \frac{1}{2} ∑_{i=1}^{m} ∇^{0}X^{i}(X^{i}) = trace_{E}[∇^{0}]^{2} + Z̄

and the X so constructed is the required map.

2.2 Torsion Skew Symmetry

A metric connection ˘∇ on the tangent bundle T M can be expanded in terms of another connection ˜∇ and its defining map X or W as introduced in Theorem 1.1.2. More precisely by (C.2) in the appendix,

˘∇_{v}U = ˜∇_{v}U + ∑_{j} ˜∇_{v}X^{j} ⟨u, X^{j}⟩ .

However this process is not reversible: we do not seem to be able to write the second term of the right hand side in terms of ˘∇ and X. For example if ˜∇ is taken to be the Levi-Civita connection, the special case when

∑_{j} ∇_{v}X^{j} ⟨U, X^{j}⟩_{x_{0}} = −\frac{1}{2}T̆(v, u),

turns out to be particularly interesting. This is just the torsion skew symmetric case. See Proposition 2.2.2 below. In this section we shall explore this situation.

A metric connection ˘∇ on T M is torsion skew symmetric if ˆ∇ is adapted to the same metric. Here is a corollary of Lemma 2.1.3.

Corollary 2.2.1 Suppose ˜∇ is a metric connection on T M. The following are equivalent: (a). ˜∇ is torsion skew symmetric; (b). the symmetric part of D̃(·, ·) vanishes; (c). ⟨T̃(u, ·), ·⟩ is skew symmetric for each u ∈ T M.

In the following we put together the equivalent conditions for a connection to be torsion skew symmetric, in terms of X.

Proposition 2.2.2 Let X be a defining map for a metric connection ˘∇ on T M as in Theorem 1.1.2. Then ˘∇ is torsion skew symmetric if and only if

∑_{i=1}^{m} ⟨X^{i}(x), v⟩ ∇_{u}X^{i} + ∑_{i=1}^{m} ⟨X^{i}(x), u⟩ ∇_{v}X^{i} = 0, or

∑_{i=1}^{m} X^{i}(x) ⟨v, ∇_{u}X^{i}⟩ + ∑_{i=1}^{m} X^{i}(x) ⟨u, ∇_{v}X^{i}⟩ = 0,

equivalently T̆(v, u) = 2 ∑_{i=1}^{m} X^{i}(x) ⟨∇_{v}X^{i}, u⟩. In this case,

∇_{v}U = ˘∇_{v}U − \frac{1}{2}T̆(v, u).  (2.2.1)

In particular, ∇_{X^{i}}X^{i} = 0 for each i.

Proof. The first identity comes from (C.4) with ˜∇ replaced by ∇:

T̆ (v, u) = ∑_{i=1}^{m} X^{i}(x) ⟨u, ∇_{v}X^{i}⟩ − ∑_{i=1}^{m} X^{i}(x) ⟨v, ∇_{u}X^{i}⟩ .  (2.2.2)

The second comes from the first and (C.3). The third is a consequence of (2.2.2) and the second. Finally (2.2.1) follows from (C.2) and the third identity.

Recall the definition of Z^{u}: Z^{u}(x) = X(x)Y (π(u))u as in §1.1.

Proposition 2.2.3 Let ˘∇ be a metric connection on T M with defining map X. In terms of the adjoint Y of X,

T̆(v_{1,}v_{2}) = X(x_{0})dY (v_{1}, v_{2}), v_{i} ∈ T_{x_{0}}M.  (2.2.3)

Furthermore the connection ˘∇ is

• the Levi-Civita connection if and only if X(x)dY (u, v) = 0 for all u, v ∈ T_{x}M, all x ∈ M, or ∇Z^{v} vanishes at x_{0} for all v ∈ T_{x_{0}}M.

• torsion skew symmetric if ∇Z^{w}|_{T_{x_{0}}M} : T_{x_{0}}M → T_{x_{0}}M is skew symmetric for all w ∈ T M, or ∇_{v}Z^{u} + ∇_{u}Z^{v} = 0 for any u, v ∈ T M, equivalently ˘∇_{U}V + ˘∇_{V} U = ∇_{U}V + ∇_{V} U for all vector fields U and V.

Proof. Recall Z^{v_{i}} = X(x)Y (x_{0})v_{i} as in (1.1.1). By Proposition 1.1.1, the definition of ˘∇,

T̆(v_{1}, v_{2}) = ˘∇_{v_{1}}Z^{v_{2}} − ˘∇_{v_{2}}Z^{v_{1}} − [Z^{v_{1}}, Z^{v_{2}}]
= X(x_{0})∇_{v_{1}}Y (v_{2}) + ∇Z^{v_{2}}(v_{1}) − X(x_{0})∇_{v_{2}}Y (v_{1}) + ∇Z^{v_{1}}(v_{2}) − [Z^{v_{1}}, Z^{v_{2}}]
= X(x_{0})∇_{v_{1}}Y (v_{2}) − X(x_{0})∇_{v_{2}}Y (v_{1}) = X(x_{0})dY (v_{1}, v_{2}).

That ∇Z^{u} vanishes for all u iff ˘∇ is the Levi-Civita connection follows from (1.1.2), the defining property of the connection.

For the equivalent conditions of torsion skew symmetricity: the first is exactly Proposition 1.3.5: ˘∇ is torsion skew symmetric if and only if

⟨v, ∇_{u}Z^{w}⟩ + ⟨u, ∇_{v}Z^{w}⟩ = 0 u, v ∈ T M.

From the equivalence of the first two identities in Proposition 2.2.2:

⟨w, ∇_{u}Z^{v}⟩ + ⟨w, ∇_{v}Z^{u}⟩ = 0.

Adding ∇_{v}U with ∇_{u}V and use (C.2) to obtain the last equivalence.

Finally if ˜∇ is a metric connection on T M we can define a differential 3-form D^{#} by

D^{#}(−, −, −) := \frac{1}{3}Alt(⟨D(−, −), −⟩) ≡ \frac{2}{3}Cyl(⟨D(−, −), −⟩) ≡ \frac{1}{3}Cyl ⟨T (−, −), −⟩ ,

where Alt is the alternating mapping. In the torsion skew symmetric case, there is a differential 3-form T^{#} from the torsion tensor:

T^{#}(u, v, w) = ⟨T (u, v), w⟩ .  (2.2.4)

Indeed there is a bijection between torsion skew symmetric connection for a given metric and 3-forms given by T ↦ T^{#} using Lemma 2.1.3. In Appendix C we see that dT^{#} and δT^{#} appear in curvature identities.

2.3 The ‘divergence operator’ δ̂

A. Let X : ℝ^{m} → T M be a smooth bundle map (not necessarily of constant rank). Let ∧^{∗}T^{∗}M be the space of differential forms on M. Define δ̂: ∧^{q}T^{∗}M → ∧^{q−1}T^{∗}M by

δ̂ϕ = − ∑_{j=1}^{m} ι_{X^{j}}L_{X^{j}}ϕ.  (2.3.1)

Here ι_{Y} ϕ is the interior product of ϕ by Y: ι_{Y} ϕ(−) = ϕ(Y, −). On smooth functions ι_{Y} = 0. In the case that X comes from an isometric immersion of M to ℝ^{m}, δ̂ is the usual divergence operator δ.

Let A be a smooth vector field on M. Consider our operator A in Hörmander form

A = \frac{1}{2} ∑ L_{X^{j}}L_{X^{j}} + L_{A}.  (2.3.2)

Since Lie differentiations also act on forms we can also extend A to operators on forms and will use A^{q} when we want to emphasize that we consider it acting on q-forms.

One of the observations which demonstrates the role of δ̂ is the following proposition:

Proposition 2.3.1

Aϕ = \frac{1}{2} ∑_{j=1}^{m} L_{X^{j}}L_{X^{j}}ϕ + L_{A}ϕ = −\frac{1}{2} (δ̂d + dδ̂) ϕ + L_{A}ϕ.  (2.3.3)

Proof. Just observe that the Lie differentiation L_{X^{j}} is given by

L_{X^{j}}ϕ = ι_{X^{j}}dϕ + d (ι_{X^{j}}ϕ)

and d commutes with the differentiation.

B. Assume that X has constant rank. The covariant derivative ˆ∇_{·}ϕ of a q-form is a linear map

ψ = ˆ∇_{·}ϕ : E → ∧^{q}T^{∗}M

over M, i.e. a section of L(E; ∧^{q}T^{∗}M). It is not obvious in the degenerate case how to apply ˆ∇ to it again: we would want

ˆ∇_{w} (ψ(U)) = ˆ∇_{w}ψ (U(x)) + ψ ( ˆ∇_{w}U) ∈ ∧^{q}T_{x}^{∗}M  (2.3.4)

for U ∈ Γ(E) and w ∈ E_{x}, but in general ˆ∇_{w}U will not lie in E_{x}. However we can use (2.3.4) to define ”tr_{E} ˆ∇_{·}ψ(·)” by

tr_{E} ˆ∇_{·}ψ(·) := ∑_{j=1}^{m} ˆ∇_{X^{j}} (ψ(X^{j}))  (2.3.5)

since ∑_{j=1}^{m} ˆ∇_{X^{j}}X^{j} = 0.

Since this agrees with (2.3.4) it will coincide with the result obtained by taking any extension ψ̃ : T M → ∧^{q}T^{∗}M and using (2.3.4) as the definition of ˜∇ψ̃, or extending ˘∇ as in Proposition 1.3.1 to some ∇^{1} on T M and using (∇^{1})′ in the usual way.

Proposition 2.3.2 Let ˘∇ be a metric connection on a subbundle E of T M and X its defining map as in Theorem 1.1.2. Then δ̂, defined by (2.3.1), does not depend on the choice of X. In fact,

δ̂ϕ(·) = −tr_{E} ˆ∇_{·}ϕ(·, ·).

Proof. Let E^{⊥} be a complementary bundle to E so that T M = E ⊕ E^{⊥}. Let ∇^{⊥} be a connection on E^{⊥} and set ∇^{1} ≡ ˘∇ ⊕ ∇^{⊥}. Observe that for a connection ˜∇ on T M with adjoint connection ˜∇′,

L_{Y} ϕ(v_{1}, …, v_{q}) = ( ˜∇_{Y} ϕ) (v_{1}, …, v_{q}) + ∑_{j=1}^{q} ϕ(v_{1}, v_{2}, …, ˜∇′_{v_{j}}Y, …, v_{q}) (2.3.6)

for v = (v_{1}, …, v_{q}) ∈ ∧^{q}T M. Take ˜∇′ to be ∇^{1}. By Proposition 1.3.1,

L_{X^{p}}ϕ(v) = ˆ∇_{X^{p}}ϕ(v) + ∑_{j=1}^{q} ϕ(v_{1}, …, ˘∇_{v_{j}}X^{p}, …, v_{q}), 1 ≤ p ≤ m,  (2.3.7)

and ∇^{⊥} is actually not involved. By the defining property of the L-W connection ˘∇,

δ̂ϕ(·) = − ∑_{1}^{m} ι_{X^{i}} ˆ∇_{X^{i}}ϕ(·) = −tr_{E} ˆ∇_{−}ϕ(−, ·).  (2.3.8)

In the Gaussian field formulation of §1.1C we have

A = \frac{1}{2}𝔼 L_{W} L_{W} + L_{A}  (2.3.9)

with δ̂ defined accordingly. The extension of (2.3.3) to this case holds in the same way as does Proposition 2.3.2. An important consequence which follows using Theorem 2.1.1 is:

Corollary 2.3.3 The operator A on forms defined by (2.3.2) or (2.3.9) depend on X or the field W only through the associated connection ˘∇ and the induced metric on E. In particular an operator A given by (2.3.9) can always be written in the form (2.3.2) using a finite set of vector fields X^{1}, …, X^{m}.

C. The rest of this section will be on the comparison of δ̂ and the usual divergence δ.

Remark 2.3.1 Let ˘∇ be a metric connection on T M with defining map X.

1. Assume ∑ ∇_{X^{j}}X^{j} = 0. Then δ̂ = δ, the usual divergence, on differential 1-

forms. (E.g. this holds in the gradient Brownian system case, or if ˘∇ is torsion skew symmetric).

2. Assume ∑_{j} ∇_{X^{j}}X^{j} = ∇h for some smooth h : M → ℝ. Then ∑_{i} L_{X^{i}}L_{X^{i}} = Δ + L_{∇h} on differential 1-forms. Here Δ is the Laplace-Beltrami operator.

3. ∑_{1}^{m} X^{i} ∧ ∇X^{i} = 0 if and only if ˘∇ = ∇. In particular for 2-forms, δ̂ = δ − ι_{∑_{i} ∇_{X^{i}}X^{i}} if and only if ˘∇ is the Levi-Civita connection.

Proof. Remarks 1 and 2 are readily seen by

δ̂ϕ(V) = − ∑_{1}^{m} ι_{X^{j}}∇_{X^{j}}ϕ − ϕ(∑_{1}^{m} ∇_{X^{j}}X^{j}).  (2.3.10)

(i.e. (2.3.6) using ∇). Similarly on differential q (> 1) forms,

\begin{aligned}δ̂ϕ(V) & = − ∑_{1}^{m} ι_{X^{j}}∇_{X^{j}}ϕ(v_{1}, …, v_{q−1}) − ϕ(∑_{1}^{m} ∇_{X^{j}}X^{j}, v_{1}, …, v_{q−1}) \\ & − ∑_{j=1}^{m} [∑_{k=1}^{q−1} ϕ(X^{j}, v_{1}, …, ∇_{v_{k}}X^{j}, …, v_{q−1})] , \end{aligned} (2.3.11)

for any V = (v_{1}, …, v_{q−1}) ∈ ∧^{q−1}T M.

For q = 2,

δ̂ϕ(V) = ∑_{1}^{m} ι_{X^{j}}∇_{X^{j}}ϕ(v_{1}) − ∑ ϕ(X^{i}, ∇_{v_{1}}X^{i}) − ∑ ϕ(∇_{X^{i}}X^{i}, v_{1}).

This leads to the second statement of Remark 3 (assuming the first). Now we show the first. Firstly ˘∇ = ∇ implies the vanishing of ∑ X^{i} ∧ ∇X^{i} by the characterization of ˘∇. Now assume ∑ X^{i} ∧ ∇X^{i} = 0, i.e. for any vectors w_{1}, w_{2},

⟨∑ X^{i} ∧ ∇X^{i}, w_{1} ∧ w_{2}⟩ = 0.

Since ∑_{j} ∇_{v}X^{j} ⟨w_{1}, X^{j}⟩ = − ∑_{j} ⟨∇_{v}X^{j}, w_{1}⟩ X^{j} by (C.3), We see that

∑_{j} ⟨w_{1}, X^{j}⟩ ⟨w_{2}, ∇_{v}X^{j}⟩ = 0

and therefore ⟨w_{2}, ∇_{v}X^{j}⟩ = 0 if X^{j} ≠ 0. This is exactly the characteristic property of the connection ˘∇ associated to X.

D. Let ˘∇ be a metric connection on E. Using the Levi-Civita connection ∇^{0} for some metric on T M extending the metric of E as in section 1.3, set

δ^{0} = − ∑_{j=1}^{m} ι_{X^{j}}∇_{X^{j}}^{0}

acting on q-forms, 1 ≤ q ≤ n, and annihilating smooth functions. So δ^{0} = δ when E = T M. Let K_{0}^{q} : ∧^{q}T^{∗}M to ∧^{q−1}T^{∗}M be defined by K_{0}^{1}ϕ ≡ 0 and for q > 1,

K_{0}^{q}ϕ = δ̂ϕ − δ^{0}ϕ + ι_{∑_{1}^{m} ∇_{X^{j}}^{0} X^{j}}(ϕ).  (2.3.12)

Then for D̂ defined by D̂(V, U) = ˆ∇_{V} U − ∇_{V}^{0} U,

K_{0}^{q}ϕ(−) = −ϕ ((dΛ)^{q}D̂(·, X^{i}) (X^{i} ∧ −)) .

Set

A^{0} = \frac{1}{2} ∑_{j=1}^{m} ∇_{X^{j}}^{0}X^{j}.

It follows from (2.3.6) and (2.3.3), for q > 1,

∑_{j=1}^{m} L_{X^{j}}L_{X^{j}}ϕ = −\frac{1}{2}(dδ^{0} + δ^{0}d)ϕ + L_{A^{0}}ϕ − \frac{1}{2}d (K_{0}^{q}ϕ) − \frac{1}{2}K_{0}^{q+1}(dϕ).  (2.3.13)

Proposition 2.3.4 For nondegenerate X, A^{q} = \frac{1}{2}Δ^{q} + L_{A^{0}} if and only if dK_{0}^{q} + K_{0}^{q+1}d = 0. Also A^{1} = \frac{1}{2}Δ + L_{A^{0}} only when ˘∇ = ∇.

Proof. The first statement of the theorem is clear. The second statement follows from part 3 of Remark 2.3.1 and K_{0}^{2}(dϕ) = − ∑_{j=1}^{m} dϕ(X^{j}, ∇_{v}X^{j}).

Note also

∑_{j} L_{X^{j}}L_{X^{j}}ϕ(v_{1}, v_{2})

= \frac{1}{2}Δ^{q}ϕ(v_{1}, v_{2}) + L_{A^{0}}ϕ(v_{1}, v_{2}) + + ∑_{1}^{m} ϕ(X^{i}, R(v_{1}, v_{2})X^{i})

+2 ∑_{1}^{m} ∇_{X^{i}}ϕ(D(v_{1}, X^{i}), v_{2}) + 2 ∑_{1}^{m} ∇_{X^{i}}ϕ(v_{1}, D(v_{2}, X^{i})) + 2ϕ(∇_{v_{1}}X^{i}, ∇_{v_{2}}X^{i}).

Finally as an example we calculate the divergence and A^{q} for the connection used in the proof of Theorem 2.1.1:

Example 2.3.5. Consider the connection constructed in the proof of Theorem 2.1.1 of §2.1. Set W = Z̄. Let ˜∇ be the associated connection. Recall

D̃(v, u) = \frac{2}{p − 1} [⟨W, u⟩ v − ⟨v, u⟩ W] .

Then − ∑_{i=1}^{m} ∇^{0}X^{i}(X^{i}) = ∑_{1}^{m} D̃(X^{i}, X^{i}) = −2W and for each j,

∑_{i=1}^{m} ϕ(X^{i}, v_{1}, …, D̃(v_{j}, X^{i}), …, v_{q−1})

= \frac{2}{p − 1} ∑_{1}^{m} ϕ (X^{i}, v_{1}, …, ⟨W, X^{i}⟩ v_{j} − ⟨v_{j}, X^{i}⟩ W, …, v_{q−1})
= \frac{2}{p − 1}ϕ (W, v_{1}, …, v_{j}, …, v_{q−1}) − \frac{2}{p − 1}ϕ (v_{j}, v_{1}, …, W, …, v_{q−1})
= \frac{4}{p − 1}ϕ (W, v_{1}, …, v_{j}, …, v_{q−1}) .

Consequently by (2.3.12), for a q-form ϕ,

δ̂ϕ = δ^{0}ϕ − ι_{∑_{i} ∇_{X^{i}}^{0}X^{i}}ϕ − \frac{4(q − 1)}{p − 1}ι_{W} ϕ = δ^{0}ϕ − [\frac{4(q − 1)}{p − 1} + 2]ι_{W} ϕ

and in the nondegenerate case

∑_{1}^{m} L_{X^{j}}L_{X^{j}} = \frac{1}{2}Δ + L_{[ \frac{4(q−1)}{n−1} +2]W} − \frac{4}{n − 1}ι_{W} d.

Thus if W is a gradient, W = ∇h say, in the non-degenerate case A^{q} restricted to closed forms is the Bismut-Witten Laplacian corresponding to the measure

exp ([\frac{8(q − 1)}{n − 1} + 4]h(x)) dx

2.4 Hörmander form generators on differential forms

We will treat in detail the case of finite dimensional noise (or equivalently a finite sum of squares of vector fields); the infinite dimensional situation can be reduced to this by Corollary 2.3.3.

Let {ξ_{t}(x)} be the solution flow to the s.d.e. (1.2.5) and P_{t} the induced semigroup on measurable forms defined by

P_{t}ϕ = 𝔼(ξ_{t})^{∗}(ϕ)χ_{t<ζ}

when the expectation exists.

Its differential generator A is given by (2.3.2) and hence (2.3.3), as seen by Itô’s formula. See e.g. [Elw92].

Proposition 2.4.1 Let M be a Riemannian manifold. Assume non-explosion and 𝔼 sup_{s≤t} |T_{x}ξ_{s}| < ∞ for each x in M. Let ϕ be a closed bounded C^{2} q-form with both δ̂ϕ and Aϕ bounded. Then P_{t}ϕ differs from ϕ by an exact form provided dP_{s}δ̂ϕ = P_{s}dδ̂ϕ, 0 ≤ s ≤ t.

Proof. The conditions allow us to apply Itô’s formula and take expectations to obtain

P_{t}ϕ = ϕ + ∫_{0}^{t} P_{s}Aϕds.

However by Theorem 2.3.1

∫_{0}^{t} P_{s}Aϕds = ∫_{0}^{t} P_{s}(dδ̂ϕ)ds
= d ∫_{0}^{t} P_{s}(δ̂ϕ)ds

as required.

Remark: The condition dP_{s} = P_{s}d holds by differentiation under the expectation sign if M is compact and somewhat more generally see [Li92], [Li94b], and [EL94]. The fact that P_{t}ϕ is cohomologous to ϕ for closed q-forms on compact M was noted in [Elw92] and on non-compact M under the related hypothesis that the flow is strongly q-complete in [Li94b]Theorem 2.4, using the fact that ∫_{σ} ξ_{t}^{∗}ϕ = ∫_{(ξ_{t})_{∗}σ} ϕ for any qsimplex σ.

Now assume X has constant rank. Then A^{0} is given by

A^{0}(f) − L_{A}f = tr ˘∇grad_{E}f = tr ˘∇df = tr ˆ∇df,

but not ˆ∇grad_{E}f in general. Here grad_{E}f is the gradient of f with respect to the metric on E induced by X:

⟨grad_{E}f, u⟩ = df(u),  any u ∈ E.  (2.4.1)

Set

A^{X} = \frac{1}{2} ∑_{1}^{m} ∇_{X^{j}}X^{j} + A.

In the nondegenerate case A^{0} = \frac{1}{2}Δ + L_{A^{X}} and for gradient Brownian systems A^{q} = \frac{1}{2}Δ+L_{A}, as can be seen in section 2.3, see also [Elw92], [Kus88]. In fact A^{q} = \frac{1}{2}Δ+L_{A}, each q, for gradient Brownian systems.

The main theorem of this section is the following Weitzenbock formula (we use the notation in Appendix B for the linear operators dΛ and δ^{2}Λ):

Theorem 2.4.2 (Weitzenbock formula) Suppose X has constant rank. Let ˘∇ be the associated connection on its image bundle E with adjoint ˆ∇, R̆ its curvature tensor and

R̆^{q}ϕ = −ϕ ((dΛ)^{q} (∑_{p=1}^{m} R̆(X^{p}, −)(X^{p})) + ∑_{p=1}^{m} δ^{2}Λ( ˘∇_{·}X^{p})(−))  (2.4.2)

the so called Weitzenbock term. Then for q > 1,

A^{q}ϕ = \frac{1}{2}tr_{E} ˆ∇_{·}( ˆ∇_{·}ϕ) + L_{A}ϕ − \frac{1}{2}R̆^{q}ϕ  (2.4.3)
= \frac{1}{2}tr_{E} ˆ∇_{·}( ˆ∇_{·}ϕ) + ˆ∇_{A}ϕ + ϕ ((dΛ)^{q}( ˘∇A)) − \frac{1}{2}R̆^{q}ϕ  (2.4.4)

and

R̆^{q}ϕ = ϕ ((dΛ)^{q} (R̆ic^{#})) + 2 ∑_{1≤i<k≤n,1≤j<l≤dim(E)} R̆_{ikjl}(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l} ϕ. (2.4.5)

Here R̆ is the curvature tensor for ˘∇ as below, with

R̆_{ikjl} = ⟨R̆(e_{i}, e_{k})e_{l}, e_{j}⟩ , 1 ≤ i, k ≤ n, 1 ≤ j, ℓ ≤ dim(E)

Also R̆ic^{#} : T M → E is defined by

R̆ic^{#}(v) = ∑_{j=1}^{m} R̆(v, X^{j}(x))X^{j}(x), v ∈ T_{x}M,

and a^{i}, (a^{i})^{∗}, 1 ≤ i ≤ n are the annihilation and creation operators corresponding to some base e_{1}, …, e_{n} of T_{x}M which extends an orthonormal base e_{1}, …, e_{dim(E)} of E_{x}.

In the nondegenerate case ⟨R̆ic^{#}(v_{1}), v_{2}⟩ is the Ricci curvature R̆ic(v_{1}, v_{2}) of ˘∇.

A. Let R̆ : T M × T M → L(E; E) be the curvature tensor for ˘∇ on E given by

R̆(U, V)W = ˘∇_{U} ( ˘∇_{V} W ) − ˘∇_{V} ( ˘∇_{U}W ) − ˘∇_{[U,V]}W

for all vector fields U, V and E-valued vector fields W. Then we have an expression for R̆ in terms of X:

R̆(u, v; z, w) := ⟨R̆(u, v)w, z⟩ = − ∑_{1}^{m} ⟨ ˘∇_{u}X^{i} ∧ ˘∇_{v}X^{i}, w ∧ z⟩ ,  (2.4.6)

See Proposition C.4.

Lemma 2.4.3 Suppose X has constant rank. For a differential q-form ϕ,

∑_{p=1}^{m} L_{X^{p}}L_{X^{p}}ϕ

= tr_{E} ˆ∇_{·}( ˆ∇_{·}ϕ) + ∑_{p=1}^{m} ϕ ((dΛ)^{q} (R̆(X^{p}, −)(X^{p}))) + ∑_{p=1}^{m} ϕ (δ^{2}Λ( ˘∇X^{p}(−))) ,

with the convention ϕ (δ^{2}Λ( ˘∇X^{p}(−))) vanishes for q = 1 and so the infinitesimal generator is given by

\begin{aligned}A^{q}ϕ & = \frac{1}{2}tr_{E} ˆ∇_{·}( ˆ∇_{·}ϕ) + L_{A}ϕ + \frac{1}{2} ∑_{p=1}^{m} ϕ ((dΛ)^{q} (R̆(X^{p}, −)(X^{p}))) \\ & +\frac{1}{2} ∑_{p=1}^{m} ϕ (δ^{2}Λ( ˘∇X^{p}(−))) . \end{aligned}  (2.4.7)

Proof. Let ϕ be a q-form, and v a q-vector. By (2.3.7),

L_{X^{p}}(L_{X^{p}}ϕ)(v) = ˆ∇_{X^{p}} ( ˆ∇_{X^{p}}ϕ) + ˆ∇_{X^{p}}ϕ ((dΛ)^{∗}( ˘∇X^{p})(v))
+ϕ ((dΛ^{∗}( ˆ∇_{X^{p}}( ˘∇X^{p}))(v)) + ˆ∇_{X^{p}}ϕ ((dΛ)^{∗}( ˘∇X^{p})(v))
+ϕ(dΛ^{∗}(∇X^{p}) ∘ dΛ^{∗}( ˘∇X^{p})(v)).

Summing up from 1 to m, the second, and the fourth term disappear by the defining property. In Appendix B setting A(−) = ˘∇_{−}X^{p} so that A^{2}(−) = ˘∇_{˘∇_{−}X^{p}}X^{p} to obtain,

∑_{p=1}^{m} L_{X^{p}}(L_{X^{p}}ϕ)(v) = tr_{E} ˆ∇_{·} ˆ∇_{·}ϕ(v) + ∑_{p=1}^{m} ϕ ((dΛ)^{∗}( ˆ∇_{X^{p}}( ˘∇X^{p}))(v))

+ ∑_{p=1}^{m} ϕ ((δ^{2}Λ)^{∗}( ˘∇X^{p})(v)) + ∑_{p=1}^{m} ϕ ((dΛ)^{∗} ( ˘∇_{˘∇X^{p}}X^{p}) (v))

Observe ∑_{p=1}^{m} ˆ∇_{X^{p}}( ˘∇_{·}X^{p})(v) = ∑_{p=1}^{m} ˘∇_{X^{p}}( ˘∇_{·}X^{p})(v), again by the defining property of the connection ˘∇.

Let U ∈ Γ(E). Then since ∑_{p=1}^{m} ˘∇_{U} ( ˘∇_{X^{p}}X^{p}) = 0,

∑_{p=1}^{m}[ ˘∇_{˘∇_{U} X^{p}}X^{p} + ˆ∇_{X^{p}}( ˘∇_{U}X^{p})] = ∑_{p=1}^{m}[ ˘∇_{˘∇_{U} X^{p}}X^{p} + ˘∇_{X^{p}}( ˘∇_{U}X^{p})]

= ∑_{p=1}^{m} [− ˘∇_{[X^{p},U]}X^{p} + ˘∇_{˘∇_{X^{p}}U}X^{p} − ˘∇_{T (X^{p},U)}X^{p}]

+ ∑_{p=1}^{m} ˘∇_{X^{p}}( ˘∇_{U}X^{p}) − ∑_{p=1}^{m} ˘∇_{U} ( ˘∇_{X^{p}}X^{p})

= ∑_{p=1}^{m} R̆(X^{p}, U)X^{p}.

The stated result now follows.

Proof of Theorem 2.4.2. Let a^{i} and (a^{i})^{∗} be respectively the annihilation and creation operators as given in Appendix B. By Corollary B.3 in the Appendix,

ϕ (δ^{2}Λ( ˘∇_{·}X^{p}(−)) = −2 ∑_{1≤i<k≤n,1≤j<l≤dim(E)}⟨ ˘∇_{e_{i}}X^{p} ∧ ˘∇_{e_{k}}X^{p}, e_{j} ∧ e_{l}⟩(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}ϕ

and by (2.4.6),

R̆_{ikjl} = ∑_{p=1}^{m} ⟨ ˘∇_{e_{i}}X^{p} ∧ ˘∇_{e_{k}}X^{p}, e_{j} ∧ e_{l}⟩ ,

giving

∑_{p=1}^{m} ϕ (δ^{2}Λ ( ˘∇_{·}X^{p}(−))) = −2 ∑_{1≤i<k≤n,1≤j<l≤dim(E)} R̆_{ikjl}(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}ϕ.

The theorem follows. Remark: The last term agrees with the term R̃_{(4)} in [CFKS87] (on page 260) if ˘∇ is the Levi-Civita connection, after applying Bianchi’s identity. Indeed: in the Levi-Civita case:

R̃_{(4)} ≡ − ∑_{i,j,k,l=1}^{n} R_{klji}(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}
= − ∑_{i,j,k,l=1}^{n} R_{ijkl}(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}
= − ∑_{1≤i<k}^{n} ∑_{j,l=1}^{n}[R_{ijkl} − R_{kjil}(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}
= − ∑_{i<k} ∑_{j<l}[R_{ijkl} − R_{kjil} − R_{ilkj} + R_{klij}](a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}.

However by Bianchi’s identity:

R_{ijkl} − R_{kjil} = −R_{kijl} = R_{ikjl},

−R_{ilkj} + R_{klij} = −R_{iklj} = R_{ikjl}

and thus

R̃_{(4)} = −2 ∑_{i<k} ∑_{j<l} R_{ikjl}(a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l}.

Note 3B In terms of the Gaussian field in the non-degenerate case the second term in R̆^{q} can be written

2𝔼 ⟨R̆(W^{(1)}, W^{(2)})W^{(3)}, W^{(4)}⟩ (a^{W (1)})^{∗}(a^{W (2)})^{∗}a^{W (3)}a^{W (4)}

where W^{(1)}, …, a^{W (4)} are independent copies of W and (a^{W (1)})^{∗} etc. are the corresponding creation and annihilation operators, e.g. (a^{W (1)})^{∗}ϕ(v) = ϕ(a^{W (1)} ∧ v).

B. Finally we give a formula for the adjoint of R̆^{q} in terms of the vector fields X^{1}, …, X^{m}, extending those given in [Elw92]. This result is to be used in section 5 for analyzing the dynamical properties of the stochastic flows. Let (R̆^{q})^{∗} : ∧^{q}T_{x}M → ∧^{q}T_{x}M be given by

ϕ ((R̆^{q})^{∗}(v)) = (R̆^{q}ϕ)(v).  (2.4.8)

Lemma 2.4.4 Let S_{t}^{i} : M → M be the solution flow of the vector fields X^{i}, i = 0, 1, …, m. Here X^{0} is taken to be A. If v_{0} ∈ ∧^{q}T_{x_{0}}M, then

∑_{1}^{m} \frac{D̂^{2}}{∂r^{2}} ∧^{q} (T S_{r}^{i}) (v_{0})|_{r=0} = − \binom{}{R̆_{x}^{q}}^{∗} (v_{0}).  (2.4.9)

Proof. In fact this follows from Itô’s formula for the flow of

dx_{t} = X(x_{t}) ∘ dB_{t} + A(x_{t})dt,

c.f. [Elw88], if ϕ is a q-form, then

ϕ(v_{T}) = ϕ(v_{0}) + ∫_{0}^{T} ∑_{1}^{m} \frac{d}{dt} [ϕ_{S_{t}^{i}(x_{r})} (∧^{q}T S_{t}^{i}(v_{r}))]|_{t=0} dB_{r}^{i}

+ \frac{1}{2} ∫_{0}^{T} ∑_{1}^{m} \frac{d^{2}}{dt^{2}} [ϕ_{S_{t}^{i}(x_{r})} (∧^{q}T S_{t}^{i}(v_{r}))]|_{t=0} dr

+ ∫_{0}^{T} ∑_{1}^{m} \frac{d}{dt} [ϕ_{S_{t}^{0}(x_{r})} (∧^{q}T S_{t}^{0}(v_{r}))]|_{t=0} dr.

From here we conclude the infinitesimal generator on q forms in terms of ˆ∇ is given by:

Aϕ(v_{0}) = \frac{1}{2} ϕ (∑_{i} \frac{D̂^{2}}{∂r^{2}} ∧^{q} T S_{r}^{i}(v_{0})|_{r=0}) + ϕ ( \frac{D̂}{∂r} ∧^{q} T S_{r}^{0}(v_{0})|_{r=0}) plus first or-

der terms and second order terms, since

\frac{d}{dt} [ϕ_{S_{t}^{i}(x_{r})} (∧^{q}T S_{t}^{i}(v_{r}))] = ˆ∇_{X^{i}(S_{t}^{i})}ϕ (∧^{q}T S_{t}^{i}(v_{r})) + ϕ (\frac{D̂}{∂t} ∧^{q} T S_{t}^{i}(v_{r})) .

Compare this with (2.3.4) to obtain the required (2.4.9).

2.5 On the infinitesimal generator

It is well known that the Laplacian is a symmetric operator. We consider the question of the self-adjointness of the operators A^{q}, coming from a general s.d.e. rather than a gradient Brownian system. We conclude, in the case of a gradient drift, A^{q} minus a zero order term is symmetric provided the associated L-W connection is torsion skew symmetric (Corollary 2.5.6).

2.5.1 Example

There is an important class of examples for which A^{q} is self adjoint although not the de Rham-Hodge (or Witten) Laplacian and here we do not even need to assume regularity of our stochastic differential equation.

Let Γ_{0} refer to C^{∞} sections with compact support.

Example 2.5.2A. Suppose the X^{e} are Killing fields for a complete Riemannian metric ⟨, ⟩^{∼} on M. Assume A ≡ 0. Let μ be the volume element for ⟨, ⟩^{∼}. Then A^{q} with domain Γ_{0}(∧^{q}T^{∗}M) using the measure μ and inner product induced by ⟨, ⟩^{∼} is symmetric.

Proof. Let ϕ, ψ ∈ Γ_{0} ∧^{q} T^{∗}M. Then

∫_{M} ⟨P_{t}ϕ, ψ⟩^{∼} dμ = ∫_{M} ⟨Eξ_{t}^{∗}ϕ(−), ψ⟩^{∼}
= 𝔼 ∫_{M} ⟨ϕ_{ξ_{t}(x)}(T_{x}ξ_{t}−, …, T_{x}ξ_{t}−), ψ_{x}(−, …, −)⟩^{∼} μ(dx).

Let f_{1}, …, f_{n} be measurable vector fields forming an orthonormal base for T_{x}M at each x of M. Our hypothesis implies that

f̃_{i}(x) := T_{ξ_{t}^{−1}(x)}ξ_{t} (f_{i} (ξ_{t}^{−1}(x))) , i = 1 to n,

also gives an orthonormal basis. (It also implies that we have a flow ξ_{·} of diffeomorphisms either from the fact that we can lift our equation to the, finite dimensional, isometry group of M, ⟨, ⟩^{∼}, see [Kun80], or by [Li94b] since T ξ_{t} consists of isometries.)

∫_{M} ⟨P_{t}ϕ, ψ⟩^{∼} dμ
= 𝔼 ∫_{M} ∑ ϕ_{ξ_{t}(x)} (T_{x}ξ_{t}(f_{ℓ_{1}}(x)), …, T_{x}ξ_{t}(f_{ℓ_{q}}(x))) ·
ψ_{x} (f_{ℓ_{1}}(x), …, f_{ℓ_{q}}(x)) μ(dx), summed over all ℓ_{1} < ℓ_{2} < … < ℓ_{q},
= 𝔼 ∫_{M} ∑ ϕ_{y} (T_{ξ_{t}^{−1}(y)}ξ_{t}(f_{ℓ_{1}}(ξ_{t}^{−1}(y))), …, T_{ξ_{t}^{−1}(y)}ξ_{t}(f_{ℓ_{q}}(ξ_{t}^{−1}(y))))
ψ_{ξ_{t}^{−1}(y)} (f_{ℓ_{1}}(ξ_{t}^{−1}(y)), …, f_{ℓ_{q}}(ξ_{t}^{−1}(y))) μ(dy)

= 𝔼 ∫_{M} ∑ ϕ_{y} (f̃_{ℓ_{1}}(y), …, f̃_{ℓ_{q}}(y)) ·
ψ_{ξ_{t}^{−1}(y)} (T ξ_{t}^{−1}f̃_{ℓ_{1}}(y), …, T ξ_{t}^{−1}f̃_{ℓ_{q}}(y)) μ(dy),

since ξ_{t} preserves μ. Now, since A ≡ 0, the law of {ξ_{t}^{−1} : 0 ≤ t < ∞} on the group of diffeomorphisms of M is the same as that of {ξ_{t} : 0 ≤ t < ∞}, [LJW84], [Kun80], [CE83]. Thus we have

∫_{M} ⟨P_{t}ϕ, ψ⟩^{∼} dμ
= 𝔼 ∫_{M} ∑ ϕ_{y} (f̃_{ℓ_{1}}(y), …, f̃_{ℓ_{q}}(y)) ψ_{ξ_{t}(y)} (T ξ_{t}f̃_{ℓ_{1}}(y), …, T ξ_{t}f̃_{ℓ_{q}}(y)) μ(dy)
= ∫_{M} ⟨ϕ, P_{t}ψ⟩^{∼} dμ

as before. Thus P_{t} is symmetric on Γ_{0}(∧^{q}T M), consequently so is its generator A^{q}, [RS80].

2.5.2 Symmetricity of the generator A^{q}

A. Consider the regular case X : \underline{ℝ}^{m} → E ⊂ T M for E a subbundle with A ∈ Γ(E). Recall the definition of tr_{E} ˆ∇_{·}ψ(·) in §2.4 C. Set

Ψϕ = ϕ ((dΛ)^{q}( ˘∇A)) − \frac{1}{2}R̆^{q}ϕ,

where R̆^{q} is as given in (2.4.5). In the following which will allow us to deduce symmetricity of A^{q} − Ψ^{q} from that of A^{0}, part IV is essentially a finite dimensional specialization of results by Bogachev &Roeckner [BR95] and by H. Long [Lon].

Theorem 2.5.1 Let Ψ(x) ∈ (∧^{q}T^{∗}M; ∧^{q}T^{∗}M) be the zero order term in (2.4.4) and μ be a Borel measure on M. Assume

(i) the s.d.e. (1.2.5) is regular with A(x) ∈ E_{x} for each x, (ii) the adjoint semi-connection ˆ∇ for (1.2.5) is adapted to a Riemannian metric ⟨, ⟩^{}˜ on T M.

Then the following are equivalent: I. The generator A^{0} with domain restricted to C_{0}^{∞}(M; R) is symmetric on L^{2}(T M). II. For a given q ∈ {1, 2, …, n − 1}, A^{q} − Ψ with domain Γ_{0} (∧^{q}T^{∗}M) is symmetric on the space of L^{2} q-forms using the measure μ and the inner product induced on ∧^{q}T^{∗}M by ⟨, ⟩^{}˜ .

Proposition 2.5.2 Under the conditions of Theorem 2.5.1, statements I and II are equivalent to each of the following: III. The adjoint of grad_{E} with domain C_{0}^{∞}(M; R) as an operator from L^{2}(M, μ; R) to L^{2} sections of E using the metric product ⟨, ⟩^{X} is given by

(grad_{E})^{∗}(U) = −tr_{E} ˘∇_{·}U − 2 ⟨A, U⟩^{X} ,

μ almost all x ∈ M, for U ∈ Γ_{0}(E). III’. For f ∈ C_{0}^{∞}(M; R) let d_{E}f be the restriction of df to E. Then as an operator from L^{2}(M, μ; R) to L^{2} section of E^{∗} using ⟨, ⟩^{X} the adjoint of d_{E} is given by

d_{E}^{∗}ϕ = δ̂ϕ − 2ϕ(A(·)).

IV. There exists a ’logarithmic derivative’ for μ

α : R^{m} → L^{0}(M; R),

a linear map of R^{m} into the space of measurable functions on M such that for each e ∈ R^{m},

∫_{M} fL_{X^{e}}g dμ = − ∫_{M} (L_{X^{e}}f + α(e)f) g dμ  (2.5.1)

all f, g smooth with compact support for fg, and A = ∑_{j=1}^{m} α(e_{j})X^{j}.

Remarks: (i). If ν = e^{2h}μ is an equivalent measure to a Borel measure μ, then the adjoint of ˆ∇ with respect to ν is given by: ˆ∇ψ + 2ψ(grad_{E}h) where ˆ∇ is the adjoint using μ.

(ii). Example 2.3.5 gives an example when A^{q} is symmetric when restricted to closed

q-forms, using the given metric on T M and the measure exp ([\frac{8(q − 1)}{n − 1} + 4]h(x)) dx.

In many geometrical situations it is the behaviour on closed forms which is important.

B. Here we give the proof of Theorem 2.5.1 and Proposition 2.5.2. We first give three lemmas.

Lemma 2.5.3 Let A_{1} be a section of E and μ be a Borel measure on M. The equation

∫_{M} (tr_{E} ˆ∇_{·}ϕ(·) + 2ϕ(A_{1})) μ(dx) = 0,  (2.5.2)

holds for all ϕ ∈ Γ_{0}(E^{∗}) if and only if for all compactly supported q-forms θ and ψ in Γ_{0}L(E, ∧^{q}T^{∗}M).

∫_{M} (tr_{E} ˆ∇_{·} (⟨ψ(·), θ⟩^{∼}) (x) + 2 ⟨ψ(A_{1})(·), θ⟩_{x}^{∼}) μ(dx) = 0  (2.5.3)

In particular if the equation (2.5.3) holds for some q ∈ {1, …, n − 1} it holds for all such q.

Proof. Clearly (2.5.3) implies (2.5.2). Now assume that (2.5.3) holds for some q. Let ϕ ∈ Γ_{0}(E^{∗}). Take X^{1}, …, X^{q} to obtain the elements of Γ_{0}L(E; ∧^{q}T^{∗}M)

ψ(·)(−) = ϕ(·) ⟨X^{1} ∧ … ∧ X^{q}, −⟩^{∼} .

Set θ(−) = ⟨X^{1} ∧ … ∧ X^{q}, −⟩^{∼} . Then (2.5.3) reduces to (2.5.2) which in turn implies (2.5.3) for all q.

Lemma 2.5.4 Let q belong to {1, …, n − 1} and A_{1} be a section of E. Assume ˆ∇ is adapted to a Riemannian metric ⟨, ⟩^{∼} on T M. Let μ be a Borel measure on M. Then the adjojnt

ˆ∇^{∗} : Γ_{0}L(E; ∧^{q}T^{∗}M) → Γ_{0}Λ^{q}T^{∗}M

of ˆ∇, in the sense of the L^{2} spaces using μ, the inner product ⟨, ⟩^{∼} on T M, ⟨, ⟩ on E and corresponding inner product on ∧^{q}T^{∗}M and L(E; ∧^{q}T^{∗}M), is given by

ˆ∇^{∗}ψ(−) = −tr_{E} ˆ∇_{·}ψ(·)(−) − 2ψ(A_{1})(−),  (2.5.4)

if and only if for all ϕ ∈ Γ_{0}(E^{∗})

∫_{M} (tr_{E} ˆ∇_{·}ϕ(·) + 2ϕ(A_{1})) μ(dx) = 0.  (2.5.5)

Proof. Let ψ̃ : T M → ∧^{q}T^{∗}M restrict to ψ on E. For θ ∈ Γ ∧^{q} T^{∗}M, consider the 1-form ψ^{θ} defined by

ψ^{θ}(v) = ⟨ψ̃(v)(−), θ(−)⟩_{x}^{∼} , v ∈ T_{x}M.

Let S_{t}^{j} denote the (possibly partial) flow of X^{j}, for some basis e_{1}, …, e_{m} of ℝ^{m} which we shall choose to be adapted at a point x_{0} of M in the sense that {e_{1}, …, e_{p}} span KerX(x_{0})^{⊥}. Recall from Lemma 1.3.4 that T_{x_{0}}S^{j} is the parallel translation ˆ//_{·} along S_{·}^{j}(x_{0}). Therefore for v ∈ T_{x_{0}}M,

⟨ ψ̃_{S_{t}^{j}(x_{0})} (T S_{t}^{j}(v)) (−), θ_{S_{t}^{j}(x_{0})}(−) ⟩_{S_{t}^{j}(x_{0})}^{∼}
= ⟨ ψ̃_{S_{t}^{j}(x_{0})} (T S_{t}^{j}(v)) (T S_{t}^{j}(−)), θ_{S_{t}^{j}(x_{0})}(T S_{t}^{j})(−) ⟩_{S_{t}^{j}(x_{0})}^{∼}
= ⟨(S_{t}^{j})^{∗} (ψ̃)(v)(−), (S_{t}^{j})^{∗} θ(−)⟩_{x_{0}}^{∼}

Thus by (2.3.1), the definition of δ̂:

−δ̂ψ^{θ}(x_{0}) = ∑_{j} ι_{X^{j}(x_{0})}L_{X^{j}} ⟨ψ̃(·)(−), θ(−)⟩^{∼}

= ∑_{j} ι_{X^{j}(x_{0})} \frac{d}{dt} ⟨(S_{t}^{j})^{∗} (ψ̃)(·)(−), (S_{t}^{j})^{∗} θ(−)⟩_{x_{0}}^{∼} |_{t=0}

= ∑_{j} ι_{X^{j}(x_{0})} {⟨L_{X^{j}}(ψ̃)(·)(−), θ(−)⟩_{x_{0}}^{∼} + ⟨ψ̃(·)(−), L_{X^{j}}θ(−)⟩_{x_{0}}^{∼}}

= ⟨∑_{j} ˆ∇_{X^{j}(x_{0})}(ψ̃)(X^{j}(x_{0}))(−), θ(−)⟩_{x_{0}}^{∼}

+ ∑_{j} ⟨ψ̃(X^{j}(x_{0}))(−), ˆ∇_{X^{j}(x_{0})}θ(−)⟩_{x_{0}}^{∼}

= ⟨tr_{E} ˆ∇_{·}ψ̃(·)(−), θ(−)⟩_{x_{0}}^{∼} + ⟨ψ_{x_{0}}, ( ˆ∇θ)_{x_{0}}⟩_{x_{0}}^{∼} .

Then for any vector field A_{1},

∫_{M} ⟨ψ, ˆ∇θ⟩^{∼} dμ = ∫_{M} {−δ̂ψ^{θ}(x) − 2ψ^{θ}(A_{1}(x))} μ(dx)
− ⟨tr_{E} ˆ∇_{·}ψ(·)(−) − 2ψ(A_{1}(x))(−), θ⟩_{x}^{∼} μ(dx).

Thus

ˆ∇^{∗}ψ = −tr_{E} ˆ∇_{·}ψ(·) − 2ψ(A_{1}(·))

if and only if for all θ,

∫_{M} (δ̂ψ^{θ} + 2ψ^{θ}(A_{1})) μ(dx) = 0.  (2.5.6)

This is (2.5.3). But, by Lemma 2.5.3, (2.5.3) is equivalent to (2.5.5).

Remark: In Lemma 2.5.4 we used two metrics and we now show this is essential unless we have torsion skew symmetricity:

Let ˜∇′ be the adjoint of a metric connection ˜∇. Let θ be a one form, then

˜∇′_{v}θ = ˜∇_{v}θ + θ(T̃(v, −)),  (2.5.7)

and

trace ˜∇′_{·}θ(·) = trace ˜∇_{·}θ(·).

So condition (2.5.5) does not change if we use the adjoint ˜∇′ replacing ˜∇. Consequently if both connections are metric connections (for the same metric) and the adjoint of one of the covariant derivatives is given in the form of (2.5.4) then so is the other. The converse is also true as seen in the following lemma:

Lemma 2.5.5 Let ˜∇ be a metric connection with ˜∇^{∗} given by

˜∇^{∗}ψ = −trace ˜∇_{·}ψ(·)(−) − 2ψ(A_{1})(−)

for 1-forms. Then the adjoint connection has ( ˜∇′)^{∗} given by

( ˜∇′)^{∗}ψ = −trace ˜∇′_{·}ψ(·)(−) − 2ψ(A_{1})(−)

if and only if ˜∇ is torsion skew symmetric.

Proof. We only need to prove that if both ˜∇^{∗} and ( ˜∇′)^{∗} are given in the prescribed form, then ˜∇′ must be a metric connection.

Without loss of generality assume A ≡ 0. Take ψ ∈ ΓL(T M, T^{∗}M). Then

tr ˜∇′ψ = ∑_{i} ˜∇_{X^{i}}ψ(X^{i}) + ∑_{i} ψ(X^{i})(T̃(X^{i}, −)).

Thus for θ a 1-form,

⟨ ˜∇′θ, ψ⟩ = − ⟨θ, tr ˜∇′_{·}ψ(·)⟩

= − ⟨θ, tr ˜∇_{·}ψ(·)⟩ − ⟨θ, ∑_{i} ψ(X^{i})(T̃(X^{i}, −))⟩ .

On the other hand

⟨ ˜∇′_{·}θ, ψ⟩ = ⟨ ˜∇_{·}θ, ψ⟩ + ⟨θ(T̃(·, −)), ψ⟩
= − ⟨θ, tr ˜∇_{·}ψ(·)⟩ + ⟨θ(T̃(·, −)), ψ⟩ ,

giving that for all such θ and ψ,

⟨θ(T̃(·, −)), ψ⟩ + ⟨θ, ∑_{i} ψ(X^{i})(T̃(X^{i}, −))⟩ = 0.  (2.5.8)

Take θ = ⟨U, −⟩ and ψ(·)(−) = ⟨V, .⟩ ⟨W, −⟩ for vector fields U, V and W. Then

⟨θ(T̃(·, −)), ψ⟩ = ∫_{M} ∑_{i,j} ⟨θ(T̃(X^{i}, X^{j}), ψ(X^{i})(X^{j})⟩ μ(dx)

= ∫_{M} ⟨U, T̃(V, W)⟩ μ(dx),

and

⟨θ, ∑_{i} ψ(X^{i})(T̃(X^{i}, −))⟩ = ∫_{M} ∑_{i,j} θ(X^{j})ψ(X^{i})(T̃(X^{i}, X^{j})) μ(dx)

= ∫_{M} ∑_{i,j} ⟨U, X^{j}⟩ ⟨V, X^{i}⟩ ⟨W, T̃(X^{i}, X^{j})⟩ μ(dx)

= ∫_{M} ⟨W, T̃(V, U)⟩ .

So

∫_{M} (⟨U, T̃(V, W)⟩ + ⟨W, T̃(V, U)⟩) μ(dx) = 0

and ˜∇ is torsion skew symmetric.

Proof of Theorem 2.5.1 and Proposition 2.5.2.

(1). Assume I. Let f ∈ C_{0}^{∞}(M; R). Then taking a suitable sequence {g_{n}} in C_{0}^{∞}(M; [0, 1]) converging to 1 we have

∫_{M} A(f)dμ = 0.

i.e.

∫_{M} (\frac{1}{2}tr_{E} ˘∇_{·}(grad_{E}f) + ⟨A, grad_{E}f⟩^{X}) dμ = 0.  (2.5.9)

For any λ ∈ C_{0}^{∞}(M; R) we see

∫_{M} \{\frac{1}{2}tr_{E} ˘∇_{·}(λgrad_{E}f) + ⟨A, λgrad_{E}f⟩^{X} dμ\}
= ∫_{M} λA^{0}(f)dμ + ∫_{M} \frac{1}{2} ⟨grad_{E}λ, grad_{E}f⟩^{X} dμ

which is symmetric in λ and f. Since ∫_{M} A^{0}(λf)dμ = 0. this gives

∫_{M} (\frac{1}{2}tr_{E} ˘∇_{·}(λ grad_{E}f) + ⟨A, λ grad_{E}f⟩^{X}) dμ = 0.

Now any U ∈ Γ_{0}(E) has the form

U(x) = ∑_{i=1}^{N} λ^{i}(x) grad_{E}f^{i}(x)

where the λ^{i} and the f^{i} are in C_{0}^{∞}(M; R); this is true when E = T M using local coordinates and a partition of unity and follows for general E by applying any projection P_{E} : T M → E. Thus (2.5.9) holds with grad_{E}f replaced by an arbitrary element of Γ_{0}(E). Replacing it by λU where λ ∈ C_{0}^{∞}(M) and U ∈ Γ_{0}(E) yields III. Thus I implies III and similarly, using A^{0} = − \frac{1}{2} δ̂d + L_{A}, we see I gives III’.

(2). Next assume III and apply it to

U(x) = f(x)X^{e}(x)

for e ∈ R^{m} and f ∈ C_{0}^{∞}(M; R) to obtain, for any g ∈ C_{0}^{∞}(M; R):

∫_{M} dg(fX^{e})dμ = − ∫_{M} g (tr_{E} ˘∇_{·}(fX^{e}) + 2 ⟨A, fX^{e}⟩^{X} dμ)
− ∫_{M} g(x) (df(X^{e}(x)) + 2 ⟨Y (x)A(x), e⟩_{R^{m}} f(x)) dμ(x)

giving (2.5.1) with

α(e)x = 2 ⟨Y (x)A(x), e⟩_{R^{m}}

and hence

A(x) = \frac{1}{2} ∑_{j} α(e_{j})X^{j}(x).

(3). Thus III gives rise to IV. Similarly III’ gives IV by applying III’ to the section of E^{∗} given by

v ↦ f(x) ⟨X^{e}(x), v⟩ :

or observe that since δ̂ = −tr ˘∇_{·} on 1-forms III and III’ are essentially the same.

(4). Next assume IV. Let ϕ ∈ Γ_{0}(E^{∗}). Then for each j, if ϕ̃ is a 1-form extending ϕ, (taking g ≡ 1)

0 = ∫_{M} {L_{X^{j}} (ϕ(X^{j})) + α(e_{j})ϕ(X^{j})} dμ
= ∫_{M} ( ˆ∇_{X^{j}} ϕ̃(X^{j}) + ϕ ( ˆ∇_{X^{j}}X^{j}) + α(e_{j})ϕ(X^{j})} dμ

giving (2.5.5) with

A = A_{1}= ∑_{j=1}^{m} α(e_{j})X^{j}.

By Lemma 2.5.4 and the Weitzenbock formula (2.4.4) we see IV implies II.

(4). To show II implies I, take f, g ∈ C_{0}^{∞}(M; R) and ϕ ∈ Γ_{0}(∧^{q}T^{∗}M). Using A = \frac{1}{2} ∑ L_{X^{j}}L_{X^{j}} + L_{A} we have

⟨A(fϕ), gϕ⟩_{L^{2}} = ∫_{M} A(f) g (|ϕ|^{}˜ )^{2} dμ + ∫_{M} fg ⟨Aϕ, ϕ⟩^{}˜ dμ

+2 ∫_{M} ∑_{j} g L_{X^{j}}f ⟨L_{X^{j}}ϕ, ϕ⟩^{}˜ dμ.

Let

Ψ(x) ∈ L (∧^{q}T^{∗}M, ∧^{q}T^{∗}M)

be the zero order term of A^{q}. Since ⟨Ψ(fϕ), gϕ⟩^{}˜ is symmetric in f, g statement II implies that this is also symmetric in f, g. If (|ϕ|^{}˜ )^{2} = 1 for x ∈ Supp g then

g ∑_{j} L_{X^{j}}f ⟨L_{X^{j}}ϕ, ϕ⟩^{}˜

vanishes identically using (2.3.6), the adaptness of ˆ∇ to ⟨, ⟩^{}˜ and the defining property of ˆ∇. In this case we therefore do have

⟨Af, g⟩_{L^{2}} = ⟨f, Ag⟩_{L^{2}.}

However we can always find an open cover {U_{i}}_{i=0}^{∞} of M with a partition of unity {λ_{i}}_{i=0}^{∞} subordinate to an open {U_{i}}_{i=0}^{∞} and with a q-form ϕ_{i} in Γ_{0}(∧^{q}T^{∗}M) with |ϕ_{i}|x̃ = 1 for x ∈ U_{i}. Then for any g ∈ C_{0}^{∞}(M; ℝ),

⟨Af, g⟩_{L^{2}} = ∑_{i} ⟨Af, λ_{i}g⟩_{L^{2}} = ∑_{i} ⟨f, A(λ_{i}g)⟩_{L^{2}} = ⟨f, Ag⟩_{L^{2}}

proving I, and completing the proof.

C. As an application we return to our stochastic differential equation assuming nondegeneracy. Let ⟨, ⟩ be the induced metric. Denote by dx the Riemannian volume measure and ∇ the Levi-Civita connection. Recall that the infinitesimal generator A^{0} for the s.d.e. on functions is symmetric if and only if the drift term A^{X} = \frac{1}{2} ∑ ∇_{X^{j}}X^{j} + A is gradient, e.g. see [IW89]. Thus in the torsion skew symmetric case, since then A^{X} = A, we only need to consider A of the form ∇h.

Corollary 2.5.6 Let h be a smooth function. Consider a nondegenerate s.d.e.:

dx_{t} = X(x_{t}) ∘ dB_{t} + ∇h(x_{t})dt.

Suppose the associated connection ˘∇ is torsion skew symmetric. Then A^{q} − Ψ is symmetric with respect to the measure e^{2h}dx for dx the volume element of the Riemannian metric induced on T M.

Example 2.5.2C. Let M = G be a Lie group and (1.2.5) a left invariant s.d.e. as in §1.3, with μ_{R} right invariant Haar measure. Then the hypotheses of the theorem are satisfied if A ≡ 0 with ⟨ ⟩^{}˜ any right invariant metric which agrees with that given on Eid by X(id). (This is a special case of Example 2.5.2A.) If left invariant Haar measure μ_{L} is used then we must take A to be the logarithmic E-derivative of the

Radon-Nikodym derivative \frac{dμ_{L}}{dμ_{R}}.

Finally we observe that if A = grad h. Then (dΛ)^{q} ˘∇A is symmetric for all q if and only if ⟨T̆(u, v), ∇h⟩ = 0, i.e. ∇h is orthogonal to the image of T. However the

R̆^{q} term is in general not symmetric. In Appendix §C the symmetricity of R̆^{1}, the Ricci curvature, is discussed.

Decomposition of noise and filtering

Throughout this chapter we will assume that we are in the non-singular situation with a subbundle E of T M and surjective vector bundle homomorphism X : \underline{ℝ}^{m} → E over M or a mean zero Gaussian field W of sections of E with the non-degeneracy assumption as in §1.1C, (which in particular allows infinite dimensional noise). In the first case, given a smooth vector field A on M there is the S.D.E.

dx_{t} = X(x_{t}) ∘ dB_{t} + A(x_{t})dt  (3.0.1)

as Example C in §1.2. We will be interested here in the case where X(x) is not injective for each x, in which case some of the noise given by {B_{t} : t ≥ 0} is ’redundant’ from the point of view of a solution x_{t} = ξ_{t}(x_{0}); t ≥ 0: we will make this notion precise, giving a decomposition of {B_{t} : t ≥ 0} into redundant and relevant noise (see Theorem 3.1.2). This will enable redundant noise to be filtered out in a variety of situations, see §3.3 below. We follow very closely the method used in [EY93] for gradient systems. We also describe the corresponding decomposition of {W_{t} : t ≥ 0} in the Gaussian field picture. As an application we show that the adjoint connection is metric with respect to some Riemannian metric for which R̆ic^{#} is bounded if and only if the derivative process and its inverse are uniformly bounded in t when conditioned on the end point.

3.1 A canonical decomposition of the noise driving a stochastic differential equation

The trivial bundle \underline{R}^{m} is decomposed into the sum of the subbundles kerX and kerX^{⊥} with total spaces ⨆{kerX(x) : x ∈ M} and ⨆{(kerX(x))^{⊥} : x ∈ M} where ⨆ refers to disjoint unions. Define the connection ˜∇ on \underline{R}^{m} to be the direct sum connection of the push forward connections induced on kerX and on (kerX)^{⊥} by the orthogonal projections K : \underline{R}^{m} → kerX and K^{⊥} : \underline{R}^{m} → (kerX)^{⊥} respectively as in §1B. Given any suitable process or path σ in M the corresponding parallel translation ˜//_{t} will preserve the decomposition of ℝ^{m} and if e ∈ [KerX (σ(0))]^{⊥} then

X(σ(t)) ˜//_{t}(e) = ˘//_{t}X (σ(0)) (e).  (3.1.1)

for the L-W connection ˘∇ on E.

Fix x_{0} in M, and define the following processes:

β_{t} := ∫_{0}^{t∧ρ} ˜//_{s}^{−1}K(x_{s})dB_{s}
B̃_{t} := ∫_{0}^{t∧ρ} ˜//_{s}^{−1}K^{⊥}(x_{s})dB_{s}
B̆_{t} := ∫_{0}^{t∧ρ} ˘//_{s}^{−1}X(x_{s})dB_{s}

where the parallel translations are along the solution {x_{s} := ξ_{s}(x_{0}) : 0 ≤ s < ρ} of (3.0.1) defined up to explosion time ρ := ρ(x_{0}). These processes are Brownian motions on kerX(x_{0}), (kerX(x_{0}))^{⊥}, and E_{x_{0}}, respectively, stopped at ρ(x_{0}), by Levy’s characterization. Note that

B̆_{t} = X(x_{0})B̃_{t}.  (3.1.2)

We first show that on [0, ρ) it is the martingale part of the stochastic anti-development of {x_{t} : 0 ≤ t < ρ} using ˘∇, (or, in case A does not have image in E, to be more precise using any connection ∇^{1} on T M extending ˘∇ as in Proposition 2A):

Lemma 3.1.1 Take a Riemannian metric on T M which extends the given one on E, giving the decomposition T M = E ⊕ E^{⊥}. Take any metric connection ∇^{⊥}, for this metric, on E^{⊥} and let ∇^{1} be the direct sum connection ˘∇ + ∇^{⊥} on T M, with parallel translations denoted by //_{·}^{1}. The corresponding stochastic anti-development

z_{t} := ∫_{0}^{t}(//_{s}^{1})^{−1} ∘ dx_{s}, 0 ≤ t < ρ  (3.1.3)

of x_{·} := ξ_{·}(x_{0}) is then given by

z_{t} := B̆_{t} + ∫_{0}^{t}(//_{s}^{1})^{−1}A(x_{s})ds, 0 ≤ t < ρ.  (3.1.4)

Proof. Since the stochastic differentials along (x_{s} : 0 ≤ s < ρ) satisfy

∘d ((//_{s}^{1})^{−1}X(x_{s})) = (//_{s}^{1})^{−1} ∘ D^{1}X(x_{s})
= (//_{s}^{1})^{−1}∇^{1}X(∘dx_{s})(−)

by definition of the covariant differential ∘D^{1}X(x_{s})(−),see [Elw88], we see that for 0 ≤ t < ρ

z_{t} = ∫_{0}^{t}(//_{s}^{1})^{−1} ∘ dx_{s} = ∫_{0}^{t}(//_{s}^{1})^{−1}X(x_{s}) ∘ dB_{s} + ∫_{0}^{t}(//_{s}^{1})^{−1}A(x_{s})ds

= ∫_{0}^{t}(//_{s}^{1})^{−1}X(x_{s})dB_{s} + \frac{1}{2}trace ∫_{0}^{t}(//_{s}^{1})^{−1}∇^{1}X (X(x_{s})−) (−)ds
+ ∫_{0}^{t}(//_{s}^{1})^{−1}A(x_{s})ds
= ∫_{0}^{t}(//_{s}^{1})^{−1}X(x_{s})dB_{s} + ∫_{0}^{t}(//_{s}^{1})^{−1}A(x_{s})ds
= B̆_{t} + ∫_{0}^{t}(//_{s}^{1})^{−1}A(x_{s})ds,

since ∇^{1}X(X(x)−) = ˘∇X(X(x)−, and by the defining properties of ˘∇ and the definition of B̆_{t}.

Recall that for any stopping time τ the σ-algebra F_{τ−} is defined to be that generated by F_{0} together with the sets {t < τ}∩A for A ∈ F_{t}. See [RY91] (4.18) (p.44). For any process {y_{s} : 0 ≤ s < ρ} let F_{t}^{y_{·}} be the σ-algebra generated by {t < ρ} together with the elements of the σ-algebra on {t < ρ} generated by y_{s}, s ≤ t, augmented as usual. Note that then F_{(t∧ρ)−}^{y_{·}} = F_{t−}^{y}. Furthermore if {τ_{n}}_{n=1}^{∞} is any increasing sequence of F_{·}^{y_{·}} -stopping times converging to ρ, then ([RY91], Ex 4.18)

F_{ρ−}^{y_{·}} = ⋁_{n} F_{τ_{n}}^{y_{·}}.  (3.1.5)

When y_{s} = ξ_{s}(x_{0}), we set F_{t}^{y_{·}} = F_{t}^{x_{0}}, etc. In the situations we will be considering we will have left continuity with F_{τ−}^{y} = F_{τ}^{y}. However we will not need to use this result and will keep to the more intuitively correct notation with the minus sign.

The following gives direct analogues of the corresponding decomposition in [EY93].

Theorem 3.1.2 (1). F B̆_{·} ; t∧ρ(x_{0})− = F_{t∧ρ(x_{0})−}^{x_{0}}, 0 ≤ t < ∞

and F B̆ ; ρ(x_{0})− = F_{ρ(x_{0})−}^{x_{0}}.

(2). If B̄_{t} = β_{t} + B̃_{t} then {B̄_{t} : t ≥ 0} is a Brownian motion on ℝ^{m} stopped at ρ(x_{0}) with

B_{t∧ρ(x_{0})} = ∫_{0}^{t∧ρ(x_{0})} ˜//_{s}dB̄_{s}.  (3.1.6)

In particular {β_{t} : t ≥ 0} and {B̆_{t} : t ≥ 0} are orthogonal martingales (and independent Brownian motions when there is no explosion).

Proof. For part (1), it is clear from (3.1.3) and (3.1.4) that F B̆_{·} ⊂ F x̆_{0}. The opposite inclusion comes from the fact that the stochastic anti-development is the inverse of the stochastic development and so is essentially known. In detail let OM denote the orthonormal frame bundle for M, with u_{0} ∈ OM a frame at x_{0} adapted to the splitting T_{x_{0}}M = E_{x_{0}} + E_{x_{0}}^{⊥} with u_{0} restricted to ℝ^{p} × {0} ⊂ ℝ^{n} identified with the restriction of X(x_{0}) to (kerX(x_{0}))^{⊥}. For u a frame at x let H_{u} : T_{x}M → T_{u}OM be the horizontal lift (using ∇^{1}). Let (x̃_{s} : 0 ≤ s < ρ} be the horizontal lift of x_{·} starting from u_{0}, so

dx̃_{s} = Hx̃_{s}X(x_{s}) ∘ dB_{s} + Hx̃_{s}A(x_{s})ds.  (3.1.7)

Using (3.1.3) and (3.1.4) and parallel translation along {x_{s} : 0 ≤ s < ρ}:

x̃_{s}u_{0}^{−1} ∘ dB̆_{s} = //_{s}^{1} ∘ dB̆_{s} = //_{s}^{1} ∘ dz_{s} − A(x_{s})ds  (3.1.8)
= ∘dx_{s} − A(x_{s})ds = X(x_{s}) ∘ dB_{s}.  (3.1.9)

Thus, if π : OM → M denotes the projection, x̃_{·} satisfies the SDE on M

dx̃_{s} = Hx̃_{s} x̃_{s}u_{0}^{−1} ∘ dB̆_{s} + Hx̃_{s}A (π(x̃_{s})) ds  (3.1.10)

which is driven by {B̆_{s} : 0 ≤ s < ∞}. Since the explosion time for x̃_{·} is ρ(x_{0}), e.g. see [Elw82] we have that ρ(x_{0}) is an F B̆_{·} ; ·-stopping time. To complete the proof of (1) it is enough to observe that, for any s.d.e. with smooth coefficients driven by a continuous martingale M_{·}, if {y_{t} : 0 ≤ t < τ(y_{0})} is a solution with explosion time τ(y_{0}) then F_{t∧τ(y_{0})}^{y_{·}} ⊂ F_{t∧τ(y_{0})}^{M_{·}}. (This is easily seen by choosing τ_{n} to be the first exit time of y_{·} from a ball of radius n about y_{0}; then τ_{n} ↗ τ(y_{0}) and F_{t∧τ_{n}}^{y_{·}} ⊂ F_{t∧τ_{n}}^{M_{·}}.)

Part (2) is immediate.

3.2 Canonical decomposition of the Gaussian field W_{t}

For the corresponding results in the Gaussian form we use the notation of §1.3C, §1.3F. We have a splitting \underline{H} = ker ρ ⊕ (ker ρ)^{⊥} of the trivial H-bundle of M with ker ρ = ⨆ {ker ρ_{x} : x ∈ M} for ρ_{x} the evaluation map at x. The projections K(·) and K^{⊥}(·) of \underline{H} onto these subbundles determine connections on the subbundles as in §1.1A and hence a direct sum connection ˜∇ on \underline{H}. The restriction of ˜∇ to (ker ρ)^{⊥} and the connection ˜∇ on E are intertwined by ρ, as therefore are the corresponding parallel translations.

There are two main complications when H is infinite dimensional. The first is that the driving process {W_{t} : t ≥ 0} is a process on Γ (E) not on H, whereas our connections is on \underline{H}. To construct the analogues of the processes β_{·}, B̃_{·} to decompose W_{·} we will therefore either have to extend the parallel translation in some sense over some space on which W_{·} lies (with no obvious choice for non-compact M) or consider the cylindrical Wiener process {W_{t}^{c} : t ≥ 0} on H, corresponding to W_{·} in the sense that if i : H → Γ(E) is the inclusion, then (as cylindrical processes) W = iW^{c}, t ≥ 0. The theory of these generalized processes and stochastic integrals with respect to them is now standard, e.g. see [Ű95] and [DPZ92] and we will use this approach. The second point is that in general solutions of linear stochastic differential equations in infinite dimensions, such as the equation for parallel transport given by a connection are not known to have versions which are almost surely linear in their initial conditions. This is more an aesthetic than a serious obstacle. However both this and the potential problems arising from the use of cylindrical processes disappear because of the finite codimensionality of ker ρ.

Indeed if u_{0} ∈ ker ρ(x_{0}) and σ : [0, ∞) → M is a smooth curve with σ(0) = x_{0} let u_{t} = /˜/_{t}u_{0} for /˜/_{t} := /˜/_{t}(σ) parallel translation along σ(t) : t ≥ 0. By definition of the connection

0 = \frac{D̃u_{t}}{∂t} = K (σ(t)) \frac{d}{dt}u_{t}

and so, since u_{t} = K (σ(t)) u_{t},

\frac{du_{t}}{dt} = (\frac{d}{dt}K (σ(t))) u_{t} = − (\frac{d}{dt}K^{⊥} (σ(t))) u_{t}.  (3.2.1)

The corresponding equation for u_{0}^{⊥} ∈ (ker ρ)^{⊥} is

\frac{du_{t}^{⊥}}{dt} = (\frac{d}{dt}K^{⊥} (σ(t))) u_{t}^{⊥}.  (3.2.2)

Now \frac{d}{dt}K^{⊥} (σ(t)) has finite rank for each t and hence is Hilbert-Schmidt, as an operator on H. From this we see that /˜/_{t} considered as a map from H to H will lie in the group O_{2}(H) which is the intersection of the orthogonal group of H with the ’Fredholm group’ GL_{2}(H):

GL_{2}(H) = {T ∈ GL(H) : T v = v + αv where α ∈ 𝕃(H; H) is Hilbert Schmidt} .

Our diffusion on M is given by

dx_{t} = ρ_{x_{t}} ∘ dW_{t} + γ̄(x_{t})dt

as in §1.1F. The parallel translation (u_{0}, u_{0}^{+}) ∈ ker ρ(x_{0}) + (ker ρ(x_{0}))^{⊥} = H along {x_{t} : 0 ≤ t < ρ} is obtained by solving the analogues of (3.2.1), (3.2.2) in their Stratonovich form. However the evolution /˜/_{t} can be obtained now as the solution of an equation on the separable Hilbert Lie group O_{2}(H), (or in the standard way by taking the horizontal lift of \{x_{t} : 0 ≤ t < ρ\} to the principal bundle of \underline{H}, taken to be the trivial O_{2}(H) bundle). Thus /˜/_{t}(x_{·}(ω)) ∈ O_{2}(H) and parallel translation on \underline{H}, and hence on the subbundle, is almost surely linear.

We can follow §3.1 above to define

β_{t}^{c} : = K(x_{0}) ∫_{0}^{t∧ρ} ˜//_{s}^{−1}dW_{s}^{c}
B̃_{t} : = ∫_{0}^{t∧ρ} /˜/_{s}^{−1}K^{⊥}(x_{s})dW_{s}^{c}
B̆_{t} : = ∫_{0}^{t∧ρ} /˘/_{s}^{−1}ρ_{x_{s}}dW_{s}.

Here β_{·}^{c} is a cylindrical Wiener process on ker ρ_{x_{0}} with β_{·} = i β^{c} a genuine Wiener process on Γ(E) since

∫_{0}^{t∧ρ} /˜/_{s}^{−1} dW_{s}^{c} = W_{t∧ρ}^{c} + ∫_{0}^{t∧ρ} λ_{s}dW_{s}^{c}

where λ_{·} is a continuous adapted Hilbert-Schmidt operator valued process, so that the stochastic integral gives a genuine continuous process in H. Note that we could also write

β_{t}^{c} = ∫_{0}^{t∧ρ} /˜/_{s}^{−1}K(x_{s})dW_{s}^{c}

to agree with §3.1 above. Since B̃_{·} and B̆_{·} take values in (ker ρ_{x_{0}})^{⊥} and E_{x_{0}} respectively they are finite dimensional. As before we can, and, will write

dW_{t}^{c} = /˜/_{t}(dβ_{t}^{c} + dB̃_{t})  (3.2.3)

observing that β_{t}^{c} + B̃_{t} is a cylindrical Wiener process on H and β_{·} and B̃_{·} are orthogonal martingales. Thus the analogue of Theorem 3.1.2 holds true in this case, and (3.2.3) is our canonical decomposition.

Remark: The real point behind this discussion is that since ker ρ has finite codimension in \underline{H} it has an induced Fredholm structure: a reduction of its structure group to GL_{2}(H^{∞−p}) where H = H^{∞−p} ⊕ H^{p} is a fixed orthogonal splitting of H with dimH^{p} = p = dimE, [ET70]; and the connection on ker ρ is compatible with this structure.

To see this, first use the fact that ker ρ is a subbundle of \underline{H} to take an open covering {u_{j} : j ∈ T }, say, of M with subbundle trivializations

θ_{j} : u_{j} → GL(H)

such that θ_{j}(x)[ker ρ_{x}] = H^{∞−p} and θ_{j}(x)[(ker ρ_{x})^{⊥}] = H^{p}. A simple argument using the fact that GL(H^{∞−p}) is open in 𝕃(H^{∞−p}; H^{∞−p}) allows us to modify the θ_{j} so as to choose them so that θ_{j}(x) differs from the identity map of H by a finite rank operator; see [ET70]. In particular θ_{j}(x) ∈ GL_{2}(H). The restrictions of the θ_{j}(x) to ker ρ give the Fredholm structure. Because of the special form of these trivializations the fact that the connection is compatible with this Fredholm structure is immediate from (3.2.1) which, as we have observed shows that parallel translates of u_{0} ∈ ker ρ_{x_{0}} differ from u_{0} by the action of Hilbert-Schmidt operators. The frame bundle of ker ρ can be taken to be an O_{2}(H^{∞−p}) bundle: in particular a separable Hilbert manifold, so that the horizontal lift of {x_{t} : 0 ≤ t < ρ} to it can be defined as usual, to give a direct version of parallel translation in ker ρ (which will of course agree with that given by (3.2.1)). From the point of view of the connection the structure group of \underline{H} is most naturally taken to be O_{2}(H^{∞−p}) × O(H^{p}) ⊂ O_{2}(H), but it will not be trivial as such a bundle in general.

3.3 Filtering out redundant noise

The following is an abstraction of the method used in [EY93] to take conditional expectations of solutions to certain linear stochastic differential equations with random coefficients. We first prove a general result: Lemma 3.3.1 on filtering of linear stochastic differential equations. It will be used in the proof of the main theorem.

By ’filtration’ we shall mean a filtration satisfying the usual conditions. Let (Ω, F, (F_{t})_{t≥0}, P) be a filtered probability space, and {N_{t}, 0 ≤ t < ∞} an R^{p}-valued continuous local F_{∗}-martingale. Denote by F_{·}^{N} the filtration generated by (N_{t}, t < ∞) and let ζ be an F_{·}^{N_{·}}-stopping time.

Let M_{·} be an R^{p}-valued continuous local F_{∗}-martingale which can be decomposed as

M_{t} = M̄_{t} + N_{t}^{⊥}  (3.3.1)

where M̄_{·} is an F_{·}^{N}-local martingale and N_{·}^{⊥} is orthogonal to N_{·} (i.e. the quadratic variation dN_{·}⊗dN_{·}^{⊥} = 0). Assume (v_{s}, 0 ≤ s < ζ) is a solution, starting from v_{0} ∈ ℝ^{n}, of the equation

dv_{s} = P_{s}(v_{s})dM_{s} + Q_{s}(v_{s})ds + R_{s}dN_{s} + r_{s}ds,  (3.3.2)

where

P : [0, ζ) × Ω → L(R^{n}, R^{p}; R^{n}),
Q : [0, ζ) × Ω → L(R^{p}; R^{n}),
R : [0, ζ) × Ω → L(R^{p}; R^{n})
and r : [0, ζ) × Ω → L(R^{p}; R^{n})

are respectively F_{∗}^{N}, F_{∗}^{N}, F_{∗}, and F_{∗} progressively measurable processes.

It turns out that the equation which governs the conditional expectation of v_{·} is of the same type as (3.3.2):

Lemma 3.3.1 Assume {N_{t}} has the predictable representation property. Let τ be a F_{·}^{N} stopping time with 0 ≤ τ < ζ such that the stopped processes P^{τ}, Q^{τ}, R^{τ}, r^{τ}, and v^{τ} satisfy

1. P^{τ} and Q^{τ} are bounded,

2. R^{τ} belongs to L^{2}(N^{τ}), (so 𝔼 ∫_{0}^{τ} |R_{s}^{τ}|^{2}d < N, N >_{s}< ∞.

3. r^{τ} belongs to L^{1} ([0, τ) × Ω),

4. v^{τ} belongs to L^{2}(M^{τ}) ∩ L^{2}(N^{τ}) ∩ L^{2} ([0, τ) × Ω).

Set

v̄_{τ∧s} = 𝔼{v_{τ∧s}|F^{N_{·}}}.

Then {v̄_{τ∧s}} is F_{·}^{N_{·}} adapted and satisfies the equation up to time τ

\{ \begin{matrix}dv̄_{s} & = P_{s}(v̄_{s})dM̄_{s} + Q_{s}(v̄_{s})ds + R̄_{s}dN_{s} + r̄_{s}ds, \\ v̄_{0} & = v_{0}, \end{matrix}  (3.3.3)

where R̄_{τ∧s} = 𝔼{R_{τ∧s}|F_{s}^{N_{·}}}, r̄_{τ∧s} = 𝔼{r_{τ∧s}|F_{s}^{N_{·}}}, and M̄ is defined in (3.3.1).

Proof. Take ϕ ∈ L^{∞}(Ω, F^{N_{·}}, P). The representation property gives an F_{·}^{N·} predictable Φ : [0, ∞) × Ω → L(R^{p}; R) with

ϕ = 𝔼ϕ + ∫_{0}^{∞} Φ_{s}(dN_{s}).

Set ϕ_{t} = 𝔼ϕ + ∫_{0}^{t} Φ_{s}(dN_{s}). Let ¯ v̄_{τ∧s} = 𝔼 {v_{τ∧s}|F_{s}^{N·}}. Note that since ϕ_{·} is both an F_{·}-martingale and an F_{·}^{N} martingale

𝔼ϕv_{τ∧t} = 𝔼ϕ_{τ∧t}v_{τ∧t} = 𝔼ϕ_{τ∧t}¯v̄_{t∧τ} = 𝔼ϕ¯v̄_{τ∧t}

so that ¯ v̄_{τ∧t} = v̄_{τ∧t}.

Next by Itô’s formula, using the orthogonality of N^{⊥} to N

ϕ_{t∧τ}v_{t∧τ} = (𝔼ϕ)v_{0} + v_{0}(ϕ_{t∧τ} − 𝔼ϕ) + (𝔼ϕ)(v_{t∧τ} − v_{0})

+ ∫_{0}^{t∧τ} Φ_{s}(dN_{s})v_{s} + ∫_{0}^{t∧τ} ϕ_{s}(P_{s}(v_{s})dM_{s} + R_{s}dN_{s})

+ ∫_{0}^{t∧τ} ϕ_{s} (Q_{s}(v_{s}) + r_{s}) ds + \frac{1}{2} ∫_{0}^{t∧τ} P_{s}(v_{s})(dM̄_{s})Φ_{s}(dN_{s})

+\frac{1}{2} ∫_{0}^{t∧τ} R_{s}(dN_{s})Φ_{s}(dN_{s}).

Then

\begin{aligned}𝔼ϕ_{t∧τ}v_{t∧τ} & = −(𝔼ϕ)v_{0} + v_{0}𝔼ϕ_{t∧τ} + (𝔼ϕ)𝔼v_{t∧τ} \\ & +𝔼 ∫_{0}^{t∧τ} ϕ_{s} (Q_{s}(v_{s}) + r̄_{s}) ds \\ & +\frac{1}{2}𝔼 ∫_{0}^{t∧τ} P_{s}(v_{s})(dM̄_{s})Φ_{s}(dN_{s}) + \frac{1}{2}𝔼 ∫_{0}^{t∧τ} R̄_{s}(dN_{s})Φ_{s}(dN_{s}) \end{aligned} (3.3.4)

using (1), (2), (3), (4). Using (1), (2), (3), (4) again, the boundedness of ϕ, and the Kunita-Watanabe inequality we obtain from (3.3.4) that

𝔼ϕ_{t∧τ} ṽ_{t∧τ} = −(Eϕ)v_{0} + v_{0}𝔼ϕ_{t∧τ} + (𝔼ϕ)𝔼v_{t∧τ} + 𝔼 ∫_{0}^{t∧τ} ϕ_{s} (Q_{s}(ṽ_{s}) + r̄_{s}) ds
+\frac{1}{2}𝔼 ∫_{0}^{t∧τ} P_{s}(ṽ_{s})(dM̄_{s})Φ_{s}(dN_{s}) + \frac{1}{2}𝔼 ∫_{0}^{t∧τ} R̄_{s}(dN_{s})Φ_{s}(dN_{s}),
= 𝔼ϕ_{t∧τ} {v_{0} + ∫_{0}^{t∧τ} P_{s}(ṽ_{s})dM̄_{s} + ∫_{0}^{t∧τ} Q_{s}(ṽ_{s})ds + ∫_{0}^{t∧τ} R̄_{s}dN_{s} + ∫_{0}^{t∧τ} r̄_{s}ds, }

(3.3.5)

whence

ṽ_{t∧τ} = v_{0} + ∫_{0}^{t∧τ} P_{s}(ṽ_{s})dM̄_{s} + ∫_{0}^{t∧τ} Q_{s}(ṽ_{s})ds + ∫_{0}^{t∧τ} R̄_{s}dN_{s} + ∫_{0}^{t∧τ} r̄_{s}ds.

To apply the previous results we will first make a general definition. Note first that if p : K → M is a vector bundle over M with possibly infinite dimensional, but separable, Banach spaces as fibres, then there are measurable trivializations θ : K → M × E where E is a linear space, with θ_{x} ≡ θ|_{p^{−1}(x)} a continuous linear isomorphism from p^{−1}(x) to E. Any two such, θ_{1}, θ_{2} say, will have θ_{1} ∘ θ_{2} : M × E → M × E measurable. Let v : Ω → K be F-measurable. Set p ∘ v = y : Ω → M and let G be some σ sub-algebra of F containing that generated by y_{·} We can define the conditional expectation of v given G by

𝔼 {v|G} = θ^{−1} (y(ω), α(ω)^{−1}𝔼 {α(·)θ_{y(·)}(v(·))|G})

whenever there exists a measurable trivialization θ and a G-measurable α : Ω → (0, ∞) such that ω ↦ α(ω)θ_{y(ω)} (v(ω)) : Ω → E is integrable. From the G-measurability of y_{·} this definition is independent of the choice of suitable trivialization; from standard results it does not depend on the choice of α: see also the proof of the next lemma. The introduction of α is helpful particularly because the trivialization does not necessarily have any relationship to any norm on E (and in practice we will want to use ones which do not).

When we have a continuous process {V_{t} : 0 ≤ t ≤ T } in K over y_{t} := pV_{t} in M we can consider ourselves to have a random variable with values in the total space of the vector bundle

C ([0, T]; K) ⟶^{p∘} C ([0, T]; M)

of continuous paths with values in K over these with values in M. When p is smooth and y_{·} is a semi-martingale (starting from a fixed y_{0} ∈ M for simplicity) any connection on p gives a parallel translation //_{t} along the paths of y_{·} and hence a measurable trivialization, almost surely defined for the law of y_{·}:

C ([0, T]; K) ⟶^{p∘} C ([0, T]; M) × C ([0, T]; p^{−1}(y_{0}))

V_{t}(ω) ↦ (y_{t} (ω) , //_{t}^{−1}V_{t} (ω)) .

When discussing conditional expectations for such processes as v_{·} it will be particularly useful to use such a trivialization, especially since in this context we will often want to take a predictable projection for some filtration {G_{t} : 0 ≤ t < ∞}. In this case we can also use localization in time to further extend the flexibility of this procedure. Moreover we also want to include processes defined only up to some stopping time:

Let {y_{t} : 0 ≤ t < τ} be a continuous process starting from a point y_{0}. Let p : K → M be a smooth vector bundle over M, possibly infinite dimensional, but separable. Let {V_{t} : 0 ≤ t < τ} be a process in K over y_{·}, i.e. pV_{t} = y_{t}. Assume τ is an F_{·}^{y_{·}}-predictable stopping time. Let G be a σ-subalgebra of F containing F_{τ−}^{y_{·}}.

Definition 3.3.2 We say that V_{·} has a local conditional expectation with respect to G, denoted by V̄_{·}, a process in K along {y_{·}}, if there exist 1. an affine connection on K (or semi-connection) with parallel translation ˜//_{t} :

T_{y_{0}}M → T_{y_{t}}M, 0 ≤ t < τ defined along {y_{t} : 0 ≤ t < τ},

2. a G-measurable process α_{·} : [0, τ) × Ω → R(> 0),

3. F_{·∧τ−}^{y_{·}}-stopping times {τ_{n} : n ≥ 0} increasing to τ such that

˜//_{t∧τ_{n}}^{−1}α_{t∧τ_{n}}V_{t∧τ_{n}} ∈ L^{1} (Ω, F, P; p^{−1}(y_{0})) ,  (3.3.6)

and

˜//_{t∧τ_{n}}𝔼{ ˜//_{t∧τ_{n}}^{−1}α_{t∧τ_{n}}V_{t∧τ_{n}}|G} = α_{t∧τ_{n}} V̄_{t∧τ_{n}}  (3.3.7)

for all t ≥ 0, n = 1, 2, . . ..

We will see the local conditional expectation is well defined in the following lemma, in the sense that if there are another set of parallel translations ˜//, function α′ and stopping times τ′_{n} increasing to τ such that the relevant random variables are integrable, then the corresponding conditional expectations must be the same:

Proposition 3.3.3 If (3.3.6) holds there is (up to modification) a unique V̄_{·} over y_{·} satisfying (3.3.7). Moreover V̄_{·} is independent of the choice of the α, the τ_{n}, and the connection satisfying (3.3.6).

Proof. For fixed n set

V̄_{t}^{n} = (α_{t∧τ_{n}})^{−1} ˜//_{t∧τ_{n}}𝔼 { ˜//_{t∧τ_{n}}^{−1}α_{t∧τ_{n}}V_{t∧τ_{n}}|G} , t ≥ 0.

For m < n we see V̄_{t∧τ_{m}}^{n} = V̄_{t}^{m}.

Then V̄_{t} is well defined up to equivalence and is similarly seen to be independent of the choice of stopping times {τ_{n}}_{n=1}^{∞} satisfying (3.3.6). Suppose now we have another set up ˜//′_{t}, α′_{t}, τ′_{n}, n = 1, 2, … as in the definition and defining V̄′_{·} by the analogue

of (3.3.7). Set τ′′_{n} = τ_{n} ∧ τ′_{n}. Since ( ˜//′_{t∧τ′′_{n}})^{−1} ˜//_{t∧τ′′_{n}} is F_{τ−}^{y} measurable, we see, from above,

α′_{t∧τ′′_{n}} V̄′_{t∧τ′′_{n}} = ˜//′_{t∧τ′′_{n}}𝔼 {( ˜//′)_{t∧τ′′_{n}}^{−1} α′_{t∧τ′′_{n}}V_{t∧τ′′_{n}}|G}
= ˜//′_{t∧τ′′_{n}} 𝔼 {( ˜//′_{t∧τ′′_{n}})^{−1} ˜//_{t∧τ′′_{n}} ˜//_{t∧τ′′_{n}}^{−1}α′_{t∧τ′′_{n}}V_{t∧τ′′_{n}}|G}
= \frac{α′_{t∧τ′′_{n}}}{α_{t∧τ′′_{n}}} ˜//_{t∧τ′′_{n}}𝔼\{//_{t∧τ′′_{n}}^{−1}α_{t∧τ′′_{n}}V_{t∧τ′′_{n}}|G\}
= α′_{t∧τ′′_{n}} V̄_{t∧τ′′_{n}}.

from above and (3.3.7). Thus V̄′_{·} = V̄_{·} as required.

Corollary 3.3.4 Suppose τ = ∞. If for some Riemannian or Finsler metric on K, ||V_{t}||_{y_{t}} ∈ L^{1} each t, then the local conditional expectation exists and is just the conditional expectation in the sense of (3.3.7).

Corollary 3.3.5 With the notation above, suppose that V_{·}, as above, has a local conditional expectation, V̄_{·} over y_{·}. Let ϕ_{t} : 0 ≤ t < τ be a G-measurable process over {y_{·}} in the dual bundle K^{∗}. If ϕ_{t}(V_{t})χ_{t<τ} is integrable then so is ϕ_{t}(V̄_{t})χ_{t<τ} and

𝔼 ϕ_{t}(V_{t})χ_{t<τ} = 𝔼 ϕ_{t}(V̄_{t})χ_{t<τ}.

Proof. With the notation of (3.3.6),

𝔼 {ϕ_{t}(V_{t})χ_{t<τ_{n}}|G} = 𝔼 {χ_{t<τ_{n}}α_{t∧τ_{n}}^{−1}ϕ_{t∧τ_{n}} ( ˜//_{t∧τ_{n}} ˜//_{t∧τ_{n}}^{−1}α_{t∧τ_{n}}V_{t∧τ_{n}}) |G}
= α_{t∧τ_{n}}^{−1}ϕ_{t∧τ_{n}} ˜//_{t∧τ_{n}}𝔼 { ˜//_{t∧τ_{n}}^{−1}α_{t∧τ_{n}}V_{t∧τ_{n}}|G} χ_{t<τ_{n}}
= ϕ_{t}(V̄_{t})χ_{t<τ_{n}}.

Thus

𝔼 sgn(ϕ_{t}(V̄_{t}))ϕ_{t}(V_{t})χ_{t<τ_{n}} = 𝔼 |ϕ_{t}(V̄_{t})|χ_{t<τ_{n}}.

Hence by the monotone convergence theorem ϕ_{t}(V̄_{t})χ_{t<τ} is integrable. Since

𝔼ϕ_{t}(V_{t})χ_{t<τ_{n}} = 𝔼ϕ_{t}(V̄_{t})χ_{t<τ_{n}}

for each n the result follows.

Remarks: (i). The independence from the choice of connections was originally pointed out by Emery. (ii). We are not assuming any Riemannian structure, and the connections are not necessarily metric for any metric.

C. The following generalizes the main result of [EY93]: we consider first a s.d.e. (3.0.1) with solution (partial) flow {ξ_{t}(x_{0}) : 0 ≤ t < ρ(x_{0}), x_{0} ∈ M}, explosion time ρ : M × Ω → (0, ∞], and derivative (partial) flow T_{x_{0}}ξ_{t} : T_{x_{0}}M → T M, x_{0} ∈ M. with the same explosion time. Assume the non-singularity condition that we have the subbundle E of T M. Recall the definition (2.4.2), (2.4.5) of the Weitzenbock curvature for the L-W connection, R̆_{x}^{q} : ∧^{q}T_{x}M → ∧^{q}T_{x}M from §2.4. First we have a crucial lemma; for the case A is not in Γ(E) see (3.3.14) below.

Lemma 3.3.6 Let 1 ≤ q ≤ n. Assume A(x) ∈ E_{x} for each x ∈ M. Let V_{0} ∈ ∧^{q}T_{x_{0}}M and V_{t} = ∧^{q}T ξ_{t}(V_{0}). Then V_{t} is a solution of the following equation (in Itô form):

D̂V_{t} = dΛ^{q} ( ˘∇X(−)dB_{t}) (V_{t}) − \frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V_{t})dt + dΛ^{q} ˘∇A(V_{t})dt, 0 ≤ t < ρ(x_{0}).

(3.3.8)

Proof. The process V_{·} satisfies the equation

D̂V_{t} = dΛ^{q} ( ˘∇X(−) ∘ dB_{t}) (V_{t}) + dΛ^{q} ˘∇A(V_{t})dt, 0 ≤ t < ρ(x_{0}),  (3.3.9)

which in Itô form (using ˆ∇) is

D̂V_{t} = dΛ^{q} ( ˘∇X(−)dB_{t}) (V_{t}) − \frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V_{t})dt + dΛ^{q} ˘∇A(V_{t})dt, 0 ≤ t < ρ(x_{0}),

(3.3.10)

since the Itô correction term for dΛ^{q} ( ˘∇X(−) ∘ dB_{t}) (V_{t}) is:

\frac{1}{2} ∑_{i=1}^{m} [dΛ^{q} ( ˆ∇_{X^{i}}( ˘∇X^{i}(−))(V_{t})) dt + dΛ^{q} ( ˘∇X^{i}( ˘∇X^{i})) (V_{t})dt
+ δ^{2}Λ^{q} ( ˘∇_{·}X^{i}) (V_{t})dt]
= \frac{1}{2} ∑_{1}^{m} [dΛ^{q} (R̆(X^{i}, ·)X^{i}) (V_{t})dt + δ^{2}Λ^{q} ( ˘∇_{·}X^{i}(V_{t}))) dt]
= −\frac{1}{2}(R̆^{q})^{∗}(V_{t})dt,

Here we used the notation from Appendix B, (2.4.7) in §2.4D and the calculations before (2.4.7).

Theorem 3.3.7 Let 1 ≤ q ≤ n. Assume A(x) ∈ E_{x} for each x ∈ M. Then the local conditional expectation of {V_{t} : 0 ≤ t < ρ(x_{0})} with respect to F_{ρ(x_{0})−}^{x_{0}} exists and is equal to the solution {V̄_{t} : 0 ≤ t < ρ(x_{0})} to

\frac{D̂V̄_{t}}{∂t} = −\frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V̄_{t}) + dΛ^{q} ˘∇A(V̄_{t}), 0 ≤ t < ρ(x_{0})  (3.3.11)

where x_{t} = ξ_{t}(x_{0}).

Proof. Using any complete metric on M let τ_{n} be the first exit time of x_{·} from the ball radius n about x_{0}, n = 1, 2 . . .. Since these balls are compact (3.3.6) holds with α_{t} ≡ 1 for any ˜//_{·}.

Now, with the notation of §1B,

˘∇X(·)dB_{s∧ρ(ω)} = ˘∇X(·) ˜//_{s}dB̄_{s} = ˘∇X(·) ˜//_{s}(dβ_{s} + dB̃_{s})
= ˘∇X(·) ˜//dβ_{s}

by the defining property of ˘∇, since ˜//_{s}β_{s} is orthogonal to KerX(x_{s}) for each s. Use ˆ//_{t}^{−1} to pull (3.3.8) back to ∧^{q}T_{y_{0}}M. We then have an equation of the form (3.3.2) for v_{s} = ˜//_{s}^{−1}V_{s}, with R ≡ 0 and M_{s} = β_{s}. Taking N = B̆_{·} and τ = τ_{n} in Lemma 3.3.1, since a stopped Brownian motion has the predictable representation property and F B̆_{·} ; ρ(x_{0})− = F_{ρ(x_{0})−}^{x_{·}} by Theorem 3.1.2, we see

𝔼{ ˆ//_{t}^{−1}V_{t∧τ_{n}}|F_{ρ(x_{0})−}^{x_{·}}} = ˆ//_{t}^{−1}V̄_{t∧τ_{n}}

for V̄_{t} given by (3.3.11), as required.

3.3.1 When A does not belong to the image of X

In the next result we give a version of (3.3.9) without the assumption that A(x) ∈ E_{x} and also translate (3.3.9) into an equation using a Levi-Civita connection. In it we keep the non-singularity condition, and the notation, of §3.3C above. We take any Riemannian metric on M and let ∇ be its Levi-Civita connection, with corresponding covariant differentiation D along x_{t} = ξ_{t}(x_{0}), 0 ≤ t < ζ(x_{0}). We also let E^{⊥} be the orthogonal bundle to E in T M with ∇^{⊥} a metric connection on E^{⊥}, and as in §1.3 set ∇^{1} = ˘∇ + ∇^{⊥} to get a connection on T M with adjoint ∇^{1′} say.

Theorem 3.3.8 For V_{t} = ∧^{q}T ξ_{t}(V_{0}) as in Theorem 3.3.7, the local conditional expectation V̄_{t}, 0 ≤ t < ρ(x_{0}), with respect to F_{ρ(x_{0})}^{x_{0}}, is adapted to F_{·}^{x_{0}} and satisfies

\frac{D^{1}′V̄_{t}}{∂t} = −\frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V̄_{t}) + dΛ^{q}(∇_{·}^{1}A)(V̄_{t})  (3.3.12)

or equivalently

DV̄_{t} = dΛ^{q} ((∇_{·}X) (Y (x_{t}) ˘//_{t} ∘ dB̆_{t})) (V̄_{t})− \frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V̄_{t})dt+dΛ^{q} (∇_{·}A) V̄_{t}dt (3.3.13)

for ˜//_{t} and B̃_{t} as in §3.1B.

Proof. We have

D^{1}′V_{t} = dΛ^{q} (∇_{·}^{1}X ∘ dB_{t}) (V_{t}) + dΛ^{q} (∇_{·}^{1}A) V_{t}dt
= dΛ^{q} ( ˘∇_{·}X ∘ dB_{t}) (V_{t}) + dΛ^{q}(∇_{·}^{1}A)V_{t}dt
= dΛ^{q} ( ˘∇_{·}XdB_{t}) (V_{t}) − \frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V_{t})dt + dΛ^{q}(∇_{·}^{1}A)V_{t}dt

by the calculations in the proof of Lemma 3.3.6, where the Itô equation is with respect to ∇^{1}. As in the proof of Theorem 3.3.7 this gives

D^{1}′V_{t} = dΛ^{q} ( ˘∇_{·}X ˜//_{t}dβ_{t}) (V_{t}) − \frac{1}{2}(R̆_{x_{t}}^{q})^{∗}(V_{t})dt + dΛ^{q}(∇_{·}^{1}A)V_{t}dt  (3.3.14)

from which the local conditional expectation {V̄_{t} : 0 ≤ t < ρ(x_{0})} exists and satisfies (3.3.12). To obtain this result in terms of ∇ it is convenient to take an X^{⊥} : \underline{R}^{ℓ} → E^{⊥} which induces ∇^{⊥} and set X^{1} = X^{⊥} + X : \underline{R}^{ℓ} ⊕ \underline{R}^{m} → T M, so that X^{1} induces ∇^{1}. By (C.8) in Proposition C.3, in Appendix C, if Y^{1} : T M → \underline{R}^{ℓ} ⊕ \underline{R}^{m} is the adjoint of X^{1} we have

D^{1}′ V̄_{t} = DV̄_{t} − dΛ^{q} (∇Z^{∘dx_{t}}) (V̄_{t})

with Z^{∘dx_{t}} = X^{1}(·)Y^{1}(x_{t})(∘dx_{t}).

By (3.1.4)

Z^{∘dx_{t}} = X^{1}(·)Y^{1}(x_{t}) (//_{t}^{1} ∘ dB̆_{t} + A(x_{t})dt)
= X(·)Y (x_{t}) ˘//_{t} ∘ dB̆_{t} + Z^{A(x_{t})}dt

Thus ∇Z^{∘dx_{t}} = ∇_{·}X (Y (x_{t}) ˘//_{t} ∘ dB̆_{t}) + ∇Z^{A(x_{t})}dt. Since, by §C

∇_{v}^{1}A = ∇_{v}A − ∇_{v}Z^{A(y)}, if v ∈ T_{y}M, y ∈ M

equation (3.3.13) follows.

We will let

W̆_{t,x_{0}}^{q,A} : ∧^{q}T_{x_{0}}M → ∧^{q}T_{x_{t}}M,  0 ≤ t < ρ(x_{0})  (3.3.15)

be the solution map determined by (3.3.11); and that given by (3.3.12) will be written

W^{1} _{t,x_{0}}^{q,A}. We will also let W_{t,x_{0}}^{q,ξ} denote the local conditional expectation of ∧^{q}T_{x_{0}}ξ_{t} with respect to F_{ρ(x_{0})−}^{x_{0}}.

For our regular s.d.e. define

\frac{ID}{∂t} ≡ \frac{ĬD^{A,q}}{∂t}

to be the operator on q-vector fields V_{t} : 0 ≤ t < ζ(x_{0}) along {ξ_{t}(x_{0}) : 0 ≤ t < ζ(x_{0})} defined by

\frac{ID}{∂t}V_{t} = \frac{D^{1}′V_{t}}{∂t} + \frac{1}{2}(R̆_{x_{t}}^{q})^{∗}V_{t} − dΛ^{q}(∇_{·}^{1}A)(V_{t})  (3.3.16)

for ∇^{1} as above.

Proposition 3.3.9 The operator \frac{ID}{∂t} and its related modified parallel transport opera-

tor W_{t,x_{0}}^{q,A}^{1} are independent of the choice of E^{⊥} and ∇^{⊥} used to define ∇^{1}.

Proof. Clearly W_{t,x_{0}}^{q,A}^{1} depends only on our s.d.e. since it is just W_{t,x_{0}}^{q,ξ}. However

\frac{ID}{∂t}V_{t} =W_{t,x_{0}}^{q,A}^{1} \frac{d}{dt} ([W_{t,x_{0}}^{q,A}^{1}]^{−1}V_{t}) .

In general \frac{ID}{∂t} will depend on ∇^{⊥} if considered as an operator on arbitrary smooth paths in M. For example if σ(t) satisfies σ̇(t) = X(σ(t))α(t) for some continuous paths α and if V is a vector field on M, setting V (t) = V (σ(t))

\frac{D^{1}′}{∂t} V (t) − ∇_{V (t)}^{1}A = ˆ∇_{σ̇(t)}V − ∇_{V (t)}^{1}A,

by Proposition 1.3.1, which will depend on ∇_{V (t)}^{1}A. (Since R̆ic depends only on ˘∇ it cannot help to make \frac{ID}{∂t} intrinsic.) However if σ̇(t) = X(σ(t))α(t) + A(σ(t)) and V (t) = V (σ(t)) then

\frac{D^{1}′}{∂t} V (t) − ∇^{1}_{V (t)}A = ˆ∇_{X(σ(t))α(t)}V + ∇^{1}′_{A(σ(t))}V − ∇_{V (t)}^{1}A
= ˆ∇_{X(σ(t))α(t)}V + [A, V](σ(t))

which is intrinsic.

3.3.2 The inverse derivative flow

In the regular case the local conditional expectation 𝔼 {Λ^{q}(T_{x_{0}}ξ_{t})^{−1}|F^{x_{0}}} is just (W_{t,x_{0}}^{q,ξ})^{−1}. Indeed using the notation of Theorem 3.3.8 to allow A(x) ∉ E_{x}.

D^{1}′ Λ^{q}(T ξ_{t})^{−1} = −Λ^{q}(T ξ_{t})^{−1} ∘ D^{1}′ (Λ^{q}(T ξ_{t})) Λ^{q}(T ξ_{t})^{−1}

whence, by (3.3.14)

D^{1}′ Λ^{q}(T_{x_{0}}ξ_{t})^{−1} = −Λ^{q}(T_{x_{0}}ξ_{t})^{−1}dΛ^{q} ( ˘∇_{·}X ˜//dB̃_{t})

+\frac{1}{2}Λ^{q}(T_{x_{0}}ξ_{t})^{−1}(R̆_{x_{t}}^{q})^{∗}dt − Λ^{q} (T ξ_{t})^{−1} dΛ^{q}(∇_{·}^{1}A)dt

from which, as before, the local conditional expectation Z_{t} : 0 ≤ t < ρ(x_{0}) is defined and satisfies

\frac{D^{1}′ Z_{t}}{∂t} = \frac{1}{2}Z_{t}(R̆_{x_{t}}^{q})^{∗} − Z_{t} dΛ^{q}(∇_{·}^{1}A)  (3.3.17)

which is just the equation for (W^{1} _{t}^{q,A})^{−1}.

3.3.3 Integrability of certain C^{r} norms for compact M

For further reference we quote the following result of Kifer with a corollary obtained as an application of the filtering result given above. Kifer’s proof was an elegant and quick application of Baxendale’s integrability theorem [Bax84]. The results could also be obtained by the direct inductive proof in [Nor86] given for certain non-compact situations. For r = 0, 1, …, and a Riemannian metric on M define

|T ξ_{t}|_{C^{r}} = sup_{x∈M} |(∇^{r}T ξ_{t})|_{x},

where ∇ is a connection on T M and T ξ_{t} is treated as a section of the bundle L(T M; ξ_{t}^{∗}(T M)) over M given its induced connection. By compactness different metrics and connections will give equivalent norms and equivalent norms would also be obtained using local coordinates systems as in [Kif88].

Proposition 3.3.10 [Kif88] For compact M and r = 0, 1, …, both sup_{0≤t≤T} |T ξ_{t}|_{C^{r}} and

sup_{0≤t≤T} |T ξ_{t}^{−1}|_{C^{r}} lie in L^{p} for 1 ≤ p < ∞, all T > 0.

Proposition 3.3.11 For a regular stochastic differential equation on a compact Rie-

mannian manifold M let W^{1} _{t,x_{0}}^{q,A} be given by (3.3.12), x_{0} ∈ M. Then both

sup_{0≤s≤T} sup_{x_{0}} |W^{1} _{t,x_{0}}^{q,A}|  and  sup_{0≤s≤T} sup_{x_{0}} |(W^{1} _{t,x_{0}}^{q,A})^{−1}|

lie in L^{p} for 1 ≤ p < ∞, all T > 0. If ˆ∇ is adapted to some metric on M they both lie in L^{∞} for each T > 0.

Proof. The case of 1 ≤ p < ∞ follows from the previous proposition and §§3.3.1, 3.3.2 above together with Jensen’s inequality for conditional expectations. When ˆ∇ is adapted to a Riemannian metric ⟨, ⟩′ choose such a metric in Theorem 3.3.8. Then by Proposition 1.3.5 the Levi-Civita covariant derivative ∇X(e) satisfies

⟨∇_{u}X(e), v⟩′ = −⟨∇_{v}X(e), u⟩′_{x}, any u, v ∈ T_{x}M, x ∈ M, e ∈ Ker^{⊥}X(x).

If we apply Itô’s formula to |V̄_{t}|_{x_{t}}^{2} in (3.3.13) we see that the Itô stochastic integral

vanishes and an L^{∞} bound for W^{1}^{q,A} follows by compactness of M. The same argument shows that there exists c ∈ ℝ such that for all v_{0} ∈ ∧^{q}T_{x_{0}}M, all x_{0} ∈ M and 0 ≤ t ≤ T

|W^{1} _{t,x_{0}}^{q,A} (V_{0})|′_{x_{t}}^{2} ≥ e^{cT} |V_{0}|_{x_{0}}^{2} a.s..

From this we obtain a uniform bound on (W^{1} _{t,x_{0}}^{q,A})^{−1}, 0 ≤ t ≤ T, x ∈ M.

A more precise result, obtained by the argument above, gives the key estimate for semigroup domination:

Proposition 3.3.12 For general M with a regular s.d.e. suppose ˆ∇ is adapted to a metric ⟨, ⟩′. Set

ρ^{q}(x) = inf{⟨R̆^{q}(V) − 2dΛ^{q}(∇A)(V), V ⟩′_{x} : V ∈ ∧^{q}T_{x}M, |V |′ ≤ 1}

where ∇ is the Levi-Civita connection for ⟨, ⟩′. Then

| W^{1} _{t,x_{0}}^{q,A} (V_{0})|′_{x_{t}} ≤ e ^{− \frac{1}{2} ∫_{0}^{t} ρ^{q}(x_{s})ds}|V_{0}|′_{x_{0}}

all V_{0} ∈ Λ^{q}T_{x_{0}}M.

3.3.4 The semigroup on forms: Bochner type vanishing theorems

For any stochastic flow the ’semi-group’ on q-forms P_{t}ϕ = 𝔼 ξ_{t}^{∗}(ϕ)χ_{t<ζ} was considered in §2.4. From Corollary 3.3.5 and Theorem 3.3.7 we know that

P_{t}ϕ(V_{0}) = 𝔼ϕ (W^{1} _{t,x_{0}}^{q,A} (V_{0})) χ_{t<ζ(x_{0})}  (3.3.18)

whenever ϕ is a q-form for which 𝔼 ξ_{t}^{∗}(ϕ)(V_{0})χ_{t<ζ(x_{0})} exists, V_{0} ∈ T_{x_{0}}M. In case ˆ∇ is adapted to some metric and ρ^{q} : M → ℝ, defined in Proposition 3.3.12, is bounded below (i.e. R^{q}− \frac{1}{2}dΛ^{q}(∇A) is bounded below) this shows that P_{t}ϕ has an ‘extension’ to a semigroup on bounded measurable forms defined by the right hand side of (3.3.18): which we will also denote by P_{t}. For future reference note Itô’s formula for ϕ(V̄_{t}) when

V̄_{t} =W^{1} _{t,x_{0}}^{q,A} (V_{0}):

\begin{aligned}ϕ(V̄_{t}) & = ϕ(V_{0}) + ∫_{0}^{t} ∇ϕ(V_{s})( ˘//_{s}dB̆_{s})(V̄_{s}) \\ & + ∫_{0}^{t} ϕ (dΛ^{q} ((∇_{·}X)(Y (x_{s}) ˘//_{s}dB̆_{s})) (V̄_{s})) \\ & + ∫_{0}^{t} A^{q}(ϕ)(V̄_{s})ds \end{aligned}  (3.3.19)

for ∇ a Levi-Civita connection.

Proposition 3.3.13 Assume nonexplosion. Suppose ˆ∇ is adapted to some metric ⟨, ⟩′ and the corresponding ρ^{q} : M → ℝ is bounded below and satisfies

lim sup_{t→∞} sup_{x∈U} 𝔼e ^{− \frac{1}{2} ∫_{0}^{t} ρ^{q}(x_{s})ds}χ_{t<ζ(x_{0})}χ_{K}(x_{t}) = 0  (3.3.20)

for all compact sets U and K of M. Then |P_{t}ϕ|′_{x} → 0 as t → ∞ for all | − |′-bounded q-forms ϕ, uniformly on compact subsets of M. In particular, if also there is no explosion, there are no non-zero bounded C^{2} forms ϕ with A^{q}ϕ = 0. If further ρ^{q−1} is bounded below and dP_{t}θ = P_{t}dθ for all compactly supported (q − 1)-forms then every closed compactly supported q-form vanishes in De Rham cohomology.

Proof. The convergence to 0 is immediate from (3.3.18) and Proposition 3.3.12 which gives the semigroup domination

|P_{t}ϕ|′_{x_{0}} ≤ 𝔼 |ϕ|′_{x_{t}} χ_{t<ζ(x_{0})} e^{− \frac{1}{2} ∫_{0}^{t} ρ^{q}(x_{s})ds}
≤ sup_{x∈M} |ϕ|′_{x} · 𝔼 e^{− \frac{1}{2} ∫_{0}^{t} ρ^{q}(x_{s})ds} χ_{t<ζ(x_{0})} χ_{supp(ϕ)}(x_{t}).  (3.3.21)

The vanishing of A^{q}-harmonic bounded forms then follows from the fact (which comes from Itô’s formula (3.3.19) and (3.3.21)) that A^{q}ϕ = 0 for ϕ bounded implies P_{t}ϕ = ϕ. Just as in the proof of Proposition 2.4.1 the commutativity of d and P_{t} implies that P_{t}ϕ is cohomologous to ϕ when dϕ = 0 using (3.3.19). However if σ is a closed qsimplex and ϕ a closed compactly supported q-form the decay of P_{t}ϕ implies the decay of ∫_{σ} P_{t}ϕ. Thus ∫_{σ} ϕ = 0 and so by De Rham’s theorem [ϕ] = 0 in H^{q}(M; ℝ).

Corollary 3.3.14 Suppose M is compact and ˆ∇ is adapted to a metric for which R̆^{q} satisfies (3.3.20). Then H^{q}(M; ℝ) = 0. In particular H^{q}(M; ℝ) = 0 if there is a subbundle E of T M and a metric connection ˘∇ on E with ˆ∇ adapted to some metric for which R̆^{q} is positive.

Proof. For M compact P_{t}ϕ = 𝔼 ξ_{t}^{∗}ϕ for bounded ϕ and dP_{t} = P_{t}d follows by differentiating under the expectation using Theorem 3.3.8. The first part then follows from the proposition and the second also using Theorem 1.1.2 to know that ˘∇ is the L-W connection for some s.d.e..

For various versions of such results and their consequences when ˘∇ is the Levi-Civita connection on T M see [ER88] and [ER91]. For relationships between Ric and R̆ic see Corollary C.7 of the Appendix and Remark (ii) following it.

3.3.5 Bismut formulae

The simplest application of the filtering procedure is to obtain Bismut formulae (in fact [EY93] originated from considering (3.3.23) below). We briefly describe a simple case, assuming that M is compact and that we have a regular s.d.e. (1.2.5) such that E is integrable and A ∈ Γ(E). Then E is preserved by T ξ_{t}. If f : M → ℝ is C^{2} and T > 0, Itô’s formula for P_{T −t}f(ξ_{t}(x_{0})) gives

f(x_{T}) = P_{T} f(x_{0}) + ∫_{0}^{T} d(P_{T −s}f)X(x_{s})dB_{s}.  (3.3.22)

Multiply both sides by ∫_{α}^{β} ⟨Y (x_{s})T ξ_{s}(v_{0}), X(x_{s})dB_{s}⟩_{x_{s}} where v_{0} ∈ E_{x_{0}} and 0 ≤ α < β ≤ T, and taking expectations giving

𝔼f(x_{T}) ∫_{α}^{β} ⟨Y (x_{s})T ξ_{s}(v_{0}), X(x_{s})dB_{s}⟩_{x_{s}} = 𝔼 ∫_{α}^{β} d(P_{T −s}f)(T ξ_{s}(v_{0}))ds
= ∫_{α}^{β} d(P_{s}(P_{T −s}f))(v_{0})ds

by differentiating under the expectation sign. Thus if v_{0} ∈ E_{x_{0}}

d(P_{T} f)(v_{0}) = \frac{1}{β − α}𝔼f(x_{T}) ∫_{α}^{β} ⟨T ξ_{s}(v_{0}), X(x_{s})dB_{s}⟩  (3.3.23)

and so by Theorem 3.3.7,

d(P_{T} f)(v_{0}) = \frac{1}{β − α}𝔼f(x_{T}) ∫_{α}^{β} ⟨W_{s}^{1}(v_{0}), ˘//_{s}dB̆_{s}⟩_{x_{s}}  (3.3.24)

since X(x_{s})dB_{s} = ˘//_{s}dB̆_{s}.

Formulae (3.3.24) now extends by continuity to continuous f : M → ℝ and exhibits the smoothing properties of P_{T} along the leaves of our foliation. From it come formulae for the logarithmic gradient of the heat kernel, proved for Brownian motions and the Levi-Civita connection by Bismut [Bis84]. For this, variations, and non-compact cases, see [EL94], [TW96] and [SZ96]. It is a primitive form of integration by parts formula like (4.1.2) below and can be proved from it (and implies it in the integrable case, as in [EL96]). Similarly (3.3.22) is an explicit form of the Clark-Ocone formula (4.1.3) below.

Application: Analysis on spaces of paths

A. For our manifold M consider C_{x_{0}} = C_{x_{0}} ([0, T], M), the space of continuous σ : [0, T] → M with σ(0) = x_{0}, equipped with the law μ = μ_{x_{0}} given by our stochastic differential equation (3.0.1).

Since C_{x_{0}} has a C^{∞} Banach manifold structure we can consider C^{1} functions F : C_{x_{0}} → ℝ. Smooth cylindrical functions are a subclass of these. There is then the (Fréchet) derivative map

dF : T C_{x_{0}} → ℝ

with

(dF)_{σ} : T_{σ}C_{x_{0}} → ℝ

a bounded linear map for each σ ∈ C_{x_{0}}.

Any Riemannian metric on M, or Finsler metric {| |_{x} : x ∈ M}, determines a Finsler metric on C_{x_{0}} with norm on T_{σ}C_{x_{0}}:

∥V_{·}∥_{σ}^{F ins} = sup_{0≤t≤T} ∥V_{t}∥_{σ(t)} .

The norms on T_{σ}C_{x_{0}} which arise this way all determine the underlying Banachable structure of T_{σ}C_{x_{0}} and are all equivalent: though not uniformly so in σ if M is not compact. We say F is BC^{1} if both F and dF ∈ L(T C_{x_{0}}; ℝ) are continuous and bounded, using such a given Finsler norm.

B. Consider first the regular case with A(x) ∈ E_{x} for each x ∈ M. Assume that there is no explosion. We shall define the ’tangent space’ for μ_{x_{0}}, relative to ˘∇, at a path σ, to be the subspace H_{σ} = H̆_{σ}^{μ} of T_{σ}C_{x_{0}} ([0, T]; M) defined for μ almost all σ by

H_{σ} := {V_{·} ∈ T_{σ}C_{x_{0}} |V_{t} = W_{t}^{A} ∫_{0}^{t} (W_{s}^{A})^{−1} ˘//_{s}ḣ_{s}ds, h ∈ L_{0}^{2,1} (E_{x_{0}})}  (4.0.1)

where the translations W_{t}^{A}, ˘//_{s} are along σ and (W_{t}^{A}) is the solution map of (3.3.11). Give it the Hilbert space structure inherited from L_{0}^{2,1} (E_{x_{0}}), so it has inner product

⟨V_{·}^{1}, V_{·}^{2}⟩_{σ} = ∫_{0}^{T} ⟨W_{t}^{A} \frac{d}{dt} [(W_{t}^{A})^{−1} V_{t}^{1}] , W_{t}^{A} \frac{d}{dt} [(W_{t}^{A})^{−1} V_{t}^{2}]⟩_{σ_{t}} dt  (4.0.2)

= ∫_{0}^{T} dt ⟨\frac{D̂}{∂t}V_{t}^{1} + \frac{1}{2}R̆ic^{#} (V_{t}^{1}) − ˘∇A (V_{t}^{1}) , \frac{D̂}{∂t}V_{t}^{2} + \frac{1}{2}R̆ic^{#} (V_{t}^{2}) − ˘∇A (V_{t}^{2})⟩

by (3.3.11). Note that if V ∈ T_{σ}C_{x_{0}} then, almost surely, V_{·} ∈ H_{σ} if and only if { ˆ//_{t}^{−1}V_{t} : 0 ≤ t ≤ T } is absolutely continuous and

\frac{D̂}{∂t}V_{t} + \frac{1}{2}R̆ic^{#} (V_{t}) − ˘∇A (V_{t}) ∈ E_{σ(t)}  (4.0.3)

for almost all t ∈ [0, T] with |V_{·}|_{σ} finite, for | |_{σ} defined by (4.0.2). Since R̆ic^{#} and ˘∇A both map T M to E, (4.0.3) can equivalently be expressed as

\frac{D̂}{∂t}V_{t} ∈ E_{σ(t)}, 0 ≤ t ≤ T,  (4.0.4)

which is a direct analogue for vector fields of the usual notion of ’horizontality’ for paths.

For a BC^{1} function F : C_{x_{0}} → ℝ the gradient ∇F := ˘∇F := ∇_{H}F is then defined as the measurable vector field, defined μ-almost surely, by ∇F (σ) ∈ H ≡ H̆_{σ}^{μ}

⟨∇F (σ) , V ⟩_{σ} = dF (V ·), V ∈ H.

C. For the regular case with A not necessarily a section of E we can define H_{σ}

as in (4.0.1) and (4.0.2) with W_{t}^{A}^{1} replacing W_{t}^{A}, using the notation of §3.3.2. The analogue of (4.0.3) will hold in the form

\frac{ID}{∂t}V_{t} ≡ \frac{D^{1}′}{∂t} V_{t} + \frac{1}{2}R̆ic^{#} (V_{t}) − ∇^{1}A (V_{t}) ∈ E_{σ(t)}  (4.0.5)

and

⟨V_{·}^{1}, V_{·}^{2}⟩_{σ} = ∫_{0}^{T} ⟨\frac{ID}{∂t}V_{t}^{1}, \frac{ID}{∂t}V_{t}^{2}⟩_{σ(t)} dt.

But in general (4.0.4) will no longer be true. Thus (4.0.5) expresses the “horizontality’ of these vector fields. Note that (4.0.5) is intrinsic by Proposition 3.3.9. The spaces H_{σ} are determined by the measure μ_{x_{0}} (i.e. by the generator A) and the choice of any metric connection on E with Riemannian metric induced by the principal symbol of A.

Our basic assumptions in this section will be

A(i) M is compact, possibly with boundary ∂M but if so A, X vanish on ∂M.
A(ii) The stochastic differential equation (1.2.5) is regular.

From Proposition 3.3.10 we have immediately

Proposition 4.0.15 Under assumptions A(i) and A(ii) the gradient ∇F of a BC^{1} function F : C_{x_{0}} → ℝ lies in L^{p} for 1 ≤ p < ∞, i.e.

∫_{C_{x_{0}}} |∇F |_{σ}^{p} dμ(σ) < ∞.

If ˆ∇ is adapted to some metric on M then |∇F | is in L^{∞}.

4.1 Integration by parts and Clark-Ocone formulae

Clark-Ocone formulae and integration by parts formulae are closely connected e.g. see [Ű95], [AM95] and [Hsu] and it will be efficient to prove them together.

A vector field V on C_{x_{0}} with V (σ) ∈ H̆_{σ}^{μ} for almost all σ will be said to be adapted if there is a version of \{\frac{ID}{∂s}V (σ)_{s} : 0 ≤ s ≤ T \} adapted to \{F_{t}^{x_{0}} : 0 ≤ t ≤ T \}. If so by \frac{ID}{∂s}V (σ)_{s} we will always mean such a version.

Theorem 4.1.1 (Integration by parts) Under assumptions A(i), A(ii) let F : C_{x_{0}} → ℝ be BC^{1} and let V be a vector field on C_{x_{0}} with V (σ) ∈ H̆_{σ}^{μ} almost surely which is adapted and has

∫_{C_{x_{0}}} |V (σ)|_{σ}^{1+ϵ} dμ_{x_{0}}(σ) < ∞.  (4.1.1)

for some ϵ > 0. Then

∫_{C_{x_{0}}} dF (V (σ)) dμ_{x_{0}}(σ) = − ∫_{C_{x_{0}}} F (σ) div_{μ}V (σ) dμ_{x_{0}}(σ)  (4.1.2)

where div_{μ}V : C_{x_{0}} → ℝ is given by

div_{μ}V (σ) = ∫_{0}^{T} ⟨W_{t}^{A}^{1} \frac{d}{dt} [(W_{t}^{A}^{1})^{−1}V (σ)_{t}] , ˘//_{t}dB̆_{t}⟩

= ∫_{0}^{T} ⟨\frac{ID}{∂t}V (σ)_{t}, ˘//_{t}dB̆_{t}⟩_{σ(t)}

where {B̆_{t}(σ), 0 ≤ t ≤ T, σ ∈ C_{x_{0}}} is the martingale part of the stochastic antidevelopment of the canonical process given by μ on C_{x_{0}}, using ˘∇. If ˆ∇ is adapted to some Riemannian metric on M we can take ϵ = 0 in (4.1.1).

Theorem 4.1.2 (Clark-Ocone formula for possibly degenerate diffusions) Let F : C_{x_{0}} → ℝ be BC^{1}. Under assumptions A(i), A(ii) for μ_{x_{0}} almost all σ ∈ C_{x_{0}}

F (σ) = ∫_{C_{x_{0}}} F (σ) dμ_{x_{0}}(σ) + ∫_{0}^{T} ⟨𝔼 \{\frac{ID}{∂t}[∇_{H}F]_{t} |F_{t}^{x_{0}} \} , ˘//_{t}dB̆_{t}⟩_{σ}  (4.1.3)

for \frac{ID}{∂t} as in (3.3.16).

Proof of Theorem 4.1.1 and 4.1.2. First we will prove Theorem 4.1.1 for a special class of V_{·}. Let 0 ≤ t_{j} < t_{j+1} ≤ T and let α_{j} be bounded ℝ^{m}-valued and F_{t_{j}}^{x_{0}}-measurable. Set k_{t} = k_{t}^{j} for

k_{t}^{j} = (t ∧ t_{j+1} − t ∧ t_{j})α_{j}

to give a bounded, F_{t}^{x_{0}}-adapted process, with paths in the Cameron-Martin space L_{0}^{2,1}([0, T]; ℝ^{m}). Let {ξ_{t}^{τ} : 0 ≤ t ≤ T } be the solution to the stochastic flow of the perturbed stochastic differential equation

dy_{t} = X(y_{t}) ∘ dB_{t} + A(y_{t})dt + τX(y_{t})k̇_{t} dt  (4.1.4)

obtained by replacing B_{·} by B_{·} + τk_{·} in (1.2.5), τ ∈ ℝ. Set x_{t}^{τ} = ξ_{t}^{τ}(x_{0}). By the Cameron-Martin theorem and Markov property of Brownian motion (or by the Girsanov-Maruyama theorem) the law of x_{·}^{τ} is equivalent to μ_{x_{0}} and

𝔼 F (x_{·}) = 𝔼F (x_{·}^{τ}) exp (−τ ∫_{0}^{T} ⟨k̇_{s}, dB_{s}⟩ − \frac{1}{2}τ^{2} ∫_{0}^{T} |k̇_{s}|^{2}ds) .

Differentiating for τ at τ = 0 under the expectation gives

𝔼 dF (v_{·}) = 𝔼F (x_{·}) ∫_{0}^{T} ⟨k̇_{s}, dB_{s}⟩  (4.1.5)

where v_{t} = \frac{∂}{∂τ} x_{t}^{τ}|_{τ=0}, 0 ≤ t ≤ T, and so satisfies

D^{1}′v_{t} = ˘∇_{v_{t}}X ∘ dB_{t} + ∇_{v_{t}}^{1}A dt + X(x_{t})k̇_{t}dt.  (4.1.6)

with v_{0} = 0. This goes back to Bismut’s approach to Malliavin’s calculus [Bis81]; see also [Nor86] where the differentiation under the expectation is carefully justified in a more general case with M not compact.

Let v̄_{t} = 𝔼{v_{t} |F_{t}^{x_{0}} }. Just as in the proof of Theorem 3.3.8 we see

\frac{D^{1}′v̄_{t}}{∂t} = −\frac{1}{2}(R̆ic_{x_{t}})(v̄_{t}) + ∇1 ; v̄_{t}A(x_{t}) + X(x_{t})(k̇_{t})  (4.1.7)

with v̄_{0} = 0, from which we see by ’variation of parameters’ that

v̄_{t} =W_{t}^{A}^{1} ∫_{0}^{t}(W_{s}^{A}^{1})^{−1}X(x_{s})k̇_{s} ds.  (4.1.8)

Set V (σ)_{t} = 𝔼{v_{t} |x_{·} = σ_{·} } so V (σ) is given by (4.1.8) with x_{·} replaced by σ. The left hand side of (4.1.5) therefore reduces to ∫_{C_{x_{0}}} dF (V (σ)) dμ_{x_{0}}(σ). For the right hand side, by Theorem 3.1.2,

𝔼 {∫_{0}^{T} ⟨k̇_{s}, dB_{s}⟩| F_{·}^{x_{0}}} = ∫_{0}^{T} ⟨k̇_{s}, ˜//_{s}dB̃_{s}⟩
= ∫_{0}^{T} ⟨k̇_{s}, Y (x_{s}) ˘//_{s}dB̆_{s}⟩
= ∫_{0}^{T} ⟨X(x_{s})k̇_{s}, ˘//_{s}dB̆_{s}⟩ .  (4.1.9)

Thus (4.1.2) holds for V of the form

V (σ)_{t} =W_{t}^{A}^{1} ∫_{0}^{t}(W_{s}^{A}^{1})^{−1}X(x_{s})k̇_{s}ds

with k = k^{j}. By linearity it holds when k is any bounded elementary ℝ^{m}-valued process, adapted to F_{·}^{x_{0}}.

Before completing the proof of Theorem 4.1.1 we will prove Theorem 4.1.2. We can assume

∫_{C_{x_{0}}} F (σ) dμ_{x}(σ) = 0.

For this let g be a bounded elementary process with values in ℝ^{m}, adapted to {F_{t} : t ≥ 0}. For some c ∈ ℝ^{m} set

G = c + ∫_{0}^{T} ⟨g_{s}, dB_{s}⟩_{ℝ^{m}} .

By the martingale representation theorem such G are dense in L^{2}(Ω, F_{T} , P; ℝ^{m}). As in (4.1.9)

𝔼{G |F_{·}^{x_{0}} } = c + ∫_{0}^{T} ⟨X(x_{s})k̇_{s}, ˘//_{s} B̆_{s}⟩

where k̇_{s} = 𝔼{g_{s} |F_{t}^{x_{0}} } is again bounded and simple. It follows from our special case of Theorem 4.1.1 that

𝔼F (x_{·})G = 𝔼 F (x_{·}) ∫_{0}^{T} ⟨X(x_{s})k̇_{s}, ˘//_{s}dB̆_{s}⟩
= 𝔼 dF (W_{·}^{A}^{1} ∫_{0}^{·}(W_{s}^{A}^{1})^{−1}X(x_{s})k̇_{s}ds)

= 𝔼 ⟨∇_{H}F, W_{·}^{A}^{1} ∫_{0}^{·}(W_{s}^{A}^{1})^{−1}X(x_{s})k̇_{s}ds⟩_{x_{·}}

= 𝔼 ∫_{0}^{T} ⟨𝔼 \{ \frac{ID}{∂t}(∇_{H}F)_{t}| F_{t}^{x_{0}}\} , X(x_{t})k̇_{t}⟩_{x_{t}} dt

= 𝔼 ∫_{0}^{T} ⟨𝔼\{\frac{ID}{∂t}(∇_{H}F)_{t} |F_{t}^{x_{0}} \}, ˘//_{t}dB̆_{t}⟩_{x_{t}} ∫_{0}^{T} ⟨X(x_{t})k̇_{t}, ˘//_{t}dB̆_{t}⟩_{x_{t}} dt

= 𝔼 ∫_{0}^{T} ⟨𝔼 \{\frac{ID}{∂t}(∇_{H}F)_{t} |F_{t}^{x_{0}} \} , ˘//_{t}dB̆_{t}⟩_{x_{t}} G,

proving (4.1.3), and Theorem 4.1.2.

To complete the proof of Theorem 4.1.1, simply multiply both sides of (4.1.3) by div_{μ}V and take expectations using Proposition 4.0.15 and (4.1.1).

Remarks: For Brownian motion measures μ these integration by parts results go back to Driver [Dri92] in the torsion skew symmetric case. As pointed out in [EL96] in the nondegenerate case our vector fields V are all “tangent processes” in the sense of Driver, for which integration by parts formulae are known see [Dri95], [CM], [AM95], and [Aid97], [Dri97b], and the monograph [Mal91] which gives further references. In the degenerate case a formula for a special class of hypoelliptic diffusions is given in [Lea].

It is shown in [EM] that from Theorem 4.1.1 follows the closability of the form

E(F, G) := ∫_{C_{x_{0}}} ⟨∇_{H}F (σ), ∇_{H}G(σ)⟩_{σ} dμ_{x_{0}}(σ)

with domain the BC^{1} functions and the result that its closure, Ē(F, G) say, is a quasi-regular local Dirichlet form on C_{x_{0}}. In particular there is an associated sample continuous process on C_{x_{0}}: the generalized Ornstein-Uhlenbeck process determined by the μ_{x_{0}} and the connection ˘∇ on the Riemannian subbundle E, ⟨, ⟩ of T M determined by μ_{x_{0}}. The general results in [EM] give an automatic extension of the integration by parts formula to a class of non-adapted vector fields with values in {H_{σ}, σ ∈ C_{x_{0}}} with an extended definition of div_{μ_{x_{0}}}.

The Clark-Ocone formula extends to F in the domain D(Ē) of Ē and immediately gives “uniqueness of the ground states” for Ē.

Corollary 4.1.3 If F ∈ D(Ē) and Ē(F, F) = 0 then F is almost surely constant. In particular if F is BC^{1} and ∇_{H}F vanishes (or equivalently dF vanishes on H_{σ}^{μ} for almost all σ) then F is almost surely constant.

We have described a family of Hilbert spaces H̆_{σ}^{μ} for each metric connection ˘∇ (and vector field A when A is not a section of E). The corollary shows that each family is sufficiently large to give at least the beginning of a Sobolev space theory. It would be interesting to know if each family is in any sense minimal with respect to the property that dF vanishes on H̆_{σ}^{μ} for almost all σ implies F almost surely constant. In [EM] the set of all tangent processes is shown to be too big to give a gradient and hence a Dirichlet form theory in any obvious way.

4.2 Logarithmic Sobolev Inequality

We can now follow the path mapped out by Capitaine-Hsu-Ledoux [CHL] for the non-degenerate case to obtain the Logarithmic Sobolev inequality for our degenerate diffusions from the Clark-Ocone formula. We include the details, based on [CHL], for completeness.

Theorem 4.2.1 Under assumptions A(i) and A(ii), the logarithmic Sobolev inequality

∫_{C_{x_{0}}} F^{2}(σ) log F^{2}(σ) dμ(σ)−∫_{C_{x_{0}}} F^{2}(σ)dμ(σ) log ∫_{C_{x_{0}}} F^{2}(σ)dμ(σ) ≤ 2 ∫_{C_{x_{0}}} |∇_{H}F |_{σ}^{2}dμ(σ)

holds for F ∈ D(Ē).

Proof. It is enough to prove it for a BC^{1} function F. Following Capitaine-Hsu-Ledoux [CHL] set

F_{t} := 𝔼 {F (ξ_{·}(x_{0})) | F_{t}^{x_{0}}}
= 𝔼 F + ∫_{0}^{t} ⟨𝔼 \{\frac{ID}{∂s}(∇_{H}F)_{s} |F_{s}^{x_{0}} \} , /˘/_{s}dB̆_{s}⟩ .

Suppose first that F > ϵ > 0. Then Itô’s formula applied to F log F gives:

𝔼(F log F) − 𝔼F log 𝔼F = \frac{1}{2}𝔼 ∫_{0}^{T} dt \frac{|𝔼 \{\frac{ID}{∂t}(∇_{H}F)_{t} |F_{t}^{x_{0}} \}|_{ξ_{t}(x_{0})}^{2}}{F_{t}} .

Replace F by F^{2} in the above and use the Cauchy-Schwartz inequality to estimate the right hand side:

|𝔼 \{\frac{ID}{∂t}(∇_{H}F^{2})_{t} |F_{t}^{x_{0}} \}|_{ξ_{t}(x_{0})}^{2} = 4 | 𝔼 \{\frac{ID}{∂t}(∇_{H}F)_{t}F |F_{t}^{x_{0}} \} |_{ξ_{t}(x_{0})}^{2}

≤ 4 𝔼 \{F^{2} |F_{t}^{x_{0}} \} 𝔼 \{ |\frac{ID}{∂t}(∇_{H}F)_{t}|^{2} |F_{t}^{x_{0}} \}.

Consequently there is the logarithmic Sobolev inequality:

∫_{C_{x_{0}}} F^{2}(σ) log F^{2}(σ) dμ(σ) − ∫_{C_{x_{0}}} F^{2}(σ)dμ(σ) log ∫_{C_{x_{0}}} F^{2}(σ)dμ(σ)

≤ 2 𝔼 ∫_{0}^{T} dt 𝔼 \{ |\frac{ID}{∂t}(∇_{H}F)_{t}|_{σ(t)}^{2} |F_{t}^{x_{0}} \}
= 2 𝔼 ∫_{0}^{T} |\frac{ID}{∂t}(∇_{H}F)_{t}|_{σ(t)}^{2} dt. = 2 ∫_{C_{x_{0}}} |∇_{H}F |_{σ}^{2} dμ(σ).

For general F ≥ 0 this holds by using (F + ϵ)^{2} instead of F^{2} etc and taking the limit.

An immediate corollary of the Logarithmic Sobolev inequality is the spectral gap inequality (e.g. see [Bak97]).

Corollary 4.2.2 For F ∈ D(Ē),

∫_{C_{x_{0}}} F^{2} dμ − (∫_{C_{x_{0}}} F dμ)^{2} ≤ \frac{1}{2}Ē(F, F).  (4.2.1)

Note that the curvature constants which have appeared in the nondegenerate case do not appear here. This is because we use a different inner product on our spaces of admissible tangent vectors, and in this case it is easy to compare these inner products when ˘∇ is metric for some metric with respect to which R̆ is bounded. However in the degenerate case we have no given Riemannian metric on M and so no canonical way of estimating curvatures, e,g, R̆ic^{#} : T M → E and W̆_{t} does not preserve E. The definition of ∇_{H} used here appears to be the most natural in the degenerate case, and so probably in the non-degenerate case.

4.3 Analysis on C_{id}(DiffM)

A. It was pointed out in [ELJL97b] that the integration by parts formula (4.1.2) was really derived from a ’mother formula’ on the space of paths on the diffeomorphism group of M. Here we give that formula together with the resulting ’mothers’ for the Clark-Ocone formula and logarithmic Sobolev inequality. As observed in [ELJL97b] the method and formulae are equally valid when the induced stochastic differential equation we use on DiffM is replaced by any right invariant systems on a Hilbert manifold with sufficiently regular group structure. We consider the Gaussian form of Proposition 1.1.3, §3.2, but use H to denote the reproducing kernel Hilbert space H_{γ} of sections of E. Recall {W_{t} : 0 ≤ t < ∞} is the Wiener process on Γ(E) which has law γ at time 1.

We assume M is compact, possibly with smooth boundary. Since the connections used here are the right and left invariant connections on DiffM no conditions on our basic stochastic differential equation on M (i.e. on the Gaussian measure γ) are assumed apart from the smoothness of the fields in H and their vanishing on ∂M.

Let K_{·}(−) ∈ L_{0}^{2,1}([0, T]; H). Let D^{s} be the Hilbert manifold of diffeomorphisms of M of Sobolev class H^{s}, s > n/2 + 3 which are the identity on ∂M. Consider the random time dependent ordinary differential equation on D^{s}, parameterized by τ ∈ R:

\frac{d}{dt}H_{t}^{τ} = τ ad(ξ_{t}^{−1})\frac{dK_{t}}{dt} , 0 ≤ t ≤ T
H_{0}^{τ} = id,

where ad(θ) denotes the adjoint action of θ ∈ DiffM on Γ(T M), i.e. ad(θ)(V) = T θ(V (θ^{−1}(·))), the push forward θ_{∗}(V). The solution exists and we can perturb our flow by it to obtain ξ_{t}^{τ} := ξ_{t} ∘ H_{t}^{τ}, 0 ≤ t ≤ T. This satisfies the analogue of (4.1.4):

dξ_{t}^{τ} = T R_{ξ_{t}^{τ}} ∘ dW_{t} + T R_{ξ_{t}^{τ}} (A)dt + τT R_{ξ_{t}^{τ}} (K̇_{t})dt.

As for x_{t}^{τ} in §4.1 the flows of ξ_{·}^{τ} on C_{id}([0, T]; DiffM) are equivalent and if F : C_{id}([0, T]; DiffM) → ℝ is bounded and measurable

𝔼 F (ξ_{·}) = 𝔼F (ξ_{·}^{τ})M^{τ}  (4.3.1)

where

M^{τ} = exp \{−τ ∫_{0}^{T} ⟨K̇_{s}, dW_{s}⟩_{H} − \frac{1}{2}τ^{2} ∫_{0}^{T} |K̇_{s}|^{2}ds\}.

To stay flexible we will say that F : C_{id}([0, T]; DiffM) → ℝ is C^{1} with suitably bounded derivative DF if one of the following holds:

Case (i), F is the restriction of a C^{1} map F : C_{id}([0, T]; D^{s}) → ℝ where the (Fréchet) derivative dF : T C_{Id}([0, T]; D^{s}) → ℝ is uniformly bounded using the Finsler metric on the tangent space T_{θ_{·}}C_{id}([0, T]; D^{s}) at θ_{·} given by

||V ||_{θ_{·}}^{F ins} := sup_{0≤t≤T} |V_{t}|_{θ_{t}}

where | · |_{θ_{t}} is the value at θ_{t} of the left invariant Riemannian structure on D^{s} determined by a standard H^{s} inner-product on Γ(T M).

Case (ii), The same as case (i) but with the Finsler metric given by

||V ||_{θ_{·}}^{F ins} := sup_{0≤t≤T} |V_{t}|_{H^{s}(θ_{t})}

where H^{s}(θ_{t}) is a standard H^{s} norm on the space of H^{s} vector fields over the diffeomorphism θ_{t}, i.e. on H^{s} sections of θ_{t}^{∗}(T M).

Case (iii), The analogue of case (i) with C^{r}DiffM replacing D^{s} for 1 ≤ r < ∞.

Case (iv), The analogue of case (ii) with C^{r}(M, M) replacing D^{s}, 0 ≤ r < ∞ or C^{r}DiffM if r ≥ 1.

We need two lemmas. The proof of the first is by straightforward calculus, since H is continuously included in C^{r}Γ(T M) for each r.

Lemma 4.3.1 Let θ : M → M be a C^{r} diffeomorphism, some r ≥ 1. Then the adjoint action ad(θ) of θ on H is continuous linear as a map ad(θ) : H → C^{r−1}Γ(T M) with norm bounded by the C^{r−1} norm of T θ.

Lemma 4.3.2 For each of the Finsler norms described above there exists Φ^{τ} : C_{id}(DiffM) → ℝ(≥ 0) with sup_{0≤τ≤1} Φ^{τ}(ξ^{τ}) in L^{p} for 1 ≤ p < ∞ such that

||T ξ_{·}^{τ} ∫_{0}^{·} ad(ξ_{s}^{τ})^{−1} K̇_{s}ds||_{ξ_{·}^{τ}}^{F ins} ≤ Φ^{τ}(ξ^{τ}) |K|_{L_{0}^{2,1}(H)}  (4.3.2)

for all K_{·} ∈ L_{0}^{2,1}(H), almost surely.

Proof. This follows from the previous lemma and Proposition 3.3.10 using the Sobolev embedding theorem to switch between C^{r} and H_{s} norms.

If F is C^{1}, bounded and with suitably bounded derivative, we can therefore differentiate (4.3.1) at τ = 0 to obtain

𝔼 dF (T ξ_{·} ∫_{0}^{·} ad(ξ_{s}^{−1})K̇_{s}ds) = 𝔼F (ξ_{·}) ∫_{0}^{T} ⟨K̇_{s}, dW_{s}⟩_{H}

which can immediately be written in terms of the law μ_{D} of ξ_{·} on DiffM, (since W_{t} = ∫_{0}^{t} T R_{ξ_{s}}^{−1} ∘ dξ_{s} − ∫_{0}^{t} T R_{ξ_{s}}^{−1}A ds))

\begin{matrix}∫_{C_{id}(DiffM)} dF (T θ_{·} ∫_{0}^{·} ad(θ_{s}^{−1}K̇_{s}ds) dμ_{D}(θ_{·}) \\ = ∫_{C_{id}(DiffM)} F (θ_{·}) ∫_{0}^{T} ⟨K̇_{s}, dW_{s}⟩_{H} dμ_{D}(θ_{·}). \end{matrix}  (4.3.3)

As we saw for C_{x_{0}}(M) this holds true when K is an adapted process with sample

paths in L_{0}^{2,1}([0, T]; H) provided that 𝔼 (∫_{0}^{T} |K̇_{s}|_{H}^{2}ds)^{\frac{1+α}{2}} < ∞ for some ϵ > 0.

C. From (4.3.3) we see that the “tangent space” we obtained for μ_{D} at θ_{·} is the Hilbert space H_{θ} = H_{θ_{·}}^{γ,A} of V_{·} ∈ T_{θ}C_{id}(DiffM) with ad(θ_{t})\frac{d}{dt} [(T θ_{t})^{−1}V_{t}] ∈ H for almost all 0 ≤ t ≤ T and having

|V_{·}|_{H_{θ}} := ∫_{0}^{T} |ad(θ_{t})\frac{d}{dt} [(T θ_{t})^{−1}V_{t}]_{H}^{2} | dt < ∞.

Note that the first condition can be written as

\frac{D̂}{dt}V_{t} ∈ T R_{θ_{t}}(H)

where now we define

\frac{D̂}{∂t}V_{t} = T θ_{t} \frac{d}{dt} [(T θ_{t})^{−1}V_{t}]

i.e. using the left invariant connection on DiffM, in complete analogy with the case of paths on M when M is a Lie group.

For our bounded C^{1} function F with suitably bounded derivative, by (4.3.2) we obtain ∇_{H}F (θ_{·}) ∈ H_{θ} for each θ, satisfying

dF (V_{·}) = ⟨∇_{H}F (θ_{·}), V_{·}⟩_{H_{θ}} , all V_{·} ∈ H_{θ}.

By (4.3.2),

|∇_{H}F (θ)|_{H(θ)} ≤ Φ(θ_{·}) ||dF ||_{θ}^{F ins}

for Φ = Φ^{0}. In particular ∇_{H}F lies in L^{p} for 1 ≤ p < ∞.

D. Just as in Theorem4.1.2 there is the Clark-Ocone formula Theorem 4.3.3 Let F : C_{id}(DiffM) → ℝ be bounded and C^{1} with suitably bounded derivative. Then

F (θ_{·}) = ∫_{C_{id}(DiffM)} F (θ_{·})dμ_{D}(θ) + ∫_{0}^{T} ⟨ad(θ_{t})𝔼 \{\frac{d}{dt} [(T θ_{t})^{−1}∇_{H}F_{t}] |F_{t} \} , dW_{t}⟩_{H}

almost surely, where F_{t} = σ{W_{s} : 0 ≤ s ≤ t}.

From this follows, as before, the logarithmic Sobolev inequality:

Theorem 4.3.4 For F bounded, C^{1}, and with suitably bounded derivative

∫_{C_{id}(DiffM)} F^{2} log F^{2}dμ_{D}(θ) − ∫_{C_{id}(DiffM)} F^{2} log ∫_{C_{id}(DiffM)} F^{2}dμ_{D}(θ)
≤ 2 ∫_{C_{id}(DiffM)} |∇_{H}F (θ)|_{H(θ)}^{2}dμ_{D}(θ).

Stability of stochastic dynamical systems

A. Consider SDE (3.0.1). Let {ξ_{t}} be the solution flow and T_{x_{0}}ξ_{t} : T_{x_{0}}M → T_{x_{0}}M the derivative flow for ξ_{t}(x_{0}). For v_{0} ∈ T_{x_{0}}M, the almost sure limit lim_{t→∞} log |T ξ_{t}(v_{0})|, called the sample Lyapunov exponent, describes the rates of convergence or divergence of solutions initiated from nearby points. We are also interested in the moment stability determined by the moment exponents:

μ_{K}(p) = lim sup_{t→∞} \frac{1}{t} log sup_{x∈K} 𝔼|T_{x}ξ_{t}|^{p}  (5.0.1)

for a subset K of M. The system (3.0.1) is strongly pth-moment stable if μ_{K}(p) < ∞ for all compact sets K. It is pth moment stable if μ_{x}(p) ≡ μ_{{x}}(p) < 0 for all x, pth moment unstable if μ_{x}(p) ≥ 0 for every x. Under suitable hypoelliptic conditions for compact manifolds, μ_{x}(p) is independent of x [BS88]. See e.g. [Elw88].

There are generalizations of the moment exponents to q-vectors:

μ_{K}^{q}(p) = lim sup_{t→∞} \frac{1}{t} log sup_{x∈K} 𝔼| ∧^{q} T_{x}ξ_{t}|^{p}  (5.0.2)

with the related concept of (q, p)-moment stability.

We shall apply the technique of filtering to obtain estimates on the moment exponents, extending that in Li [Li94a] for gradient systems, (with corresponding homotopy vanishing result extending Elworthy-Rosenberg [ER96]). We also use the L-W connection to give a neat form to a Carverhill’s version of Khasminskii’s formula, and show that in certain situations an L^{∞} condition on the derivative flow implies that ˆ∇ is metric form some metric on M.

B. Assume that X has constant rank and image E ⊂ T M with ˘∇ the associated L-W connection for X. Write E = Im(X) and define H_{p}^{q} : ∧^{q}E → ∧^{q}E by

\begin{aligned}H_{p}^{q}(V, V) = & ∑_{i=1}^{m} \frac{1}{|V |^{′2}} |dΛ^{q}( ˘∇X^{i})(V)|^{′2} \\ & +(p − 2) ∑_{1}^{m} \frac{1}{|V |^{′4}} ⟨V, dΛ^{q}( ˘∇X^{i})(V)⟩^{′2} \\ & +\frac{1}{|V_{s}|^{′2}} ⟨V, −(R̆_{x}^{q})^{∗}(V)⟩′ + 2 ∫_{0}^{t} \frac{⟨(dΛ^{q} ˘∇A)(V),V ⟩}{|V |^{′2}}^{′2}ds. \end{aligned}  (5.0.3)

Let P_{x}^{q} be the set of primitive vectors in ∧^{q}E_{x} and set

\underline{h}_{p}^{q}(x) = inf\{ p H_{p}^{q}(V, V) : V ∈ P_{x}^{q}, |V |′ = 1\}

and

h_{p}^{q}(x) = sup{ p H_{p}^{q}(V, V) : V ∈ P_{x}^{q}, |V |′ = 1}.

Then,

Theorem 5.0.5 Assume the stochastic differential equation does not explode, A(x) ∈ E_{x} for each x, and that ˆ∇ is metric with respect to a metric ⟨, ⟩′ on T M.

Then for p ∈ R, V_{0} ∈ P_{x_{0}}^{q} and V_{t} = ∧^{q}T ξ_{t}(V_{0}),

|V_{0}|^{′p} 𝔼 exp (\frac{1}{2} ∫_{0}^{t} \underline{h}_{p}^{q}(x_{s})ds) ≤ 𝔼 |V_{t}|^{′p} ≤ |V_{0}|^{′p} 𝔼 exp (\frac{1}{2} ∫_{0}^{t} h_{p}^{q}(x_{s})ds) . (5.0.4)

Here the norm | − |′ corresponds to the metric ⟨, ⟩′.

Proof. First note that by (3.3.9) D̂V_{t} ∈ ∧^{q}E. Also ∧^{q}T ξ_{t} maps primitive vectors to primitive vectors. By Itô’s formula and Lemma 3.3.6,

\begin{aligned}|V_{T} |^{′p} = & |V_{0}|^{′p} + ∑_{1}^{m} p ∫_{0}^{T} |V_{s}|^{′p−2} ⟨V_{s}, dΛ^{q} ( ˘∇X(−)dB_{s}) (V_{s})⟩′ \\ & +\frac{p}{2} ∫_{0}^{T} |V_{s}|^{′p}H̄_{p}^{q}(V_{s}, V_{s})ds, \end{aligned}  (5.0.5)

where

\begin{aligned}H̄_{p}^{q}(V, V) = & ∑_{i=1}^{m} \frac{1}{|V |^{′2}} |dΛ^{q}( ˘∇X^{i})(V)|^{′2} \\ & +(p − 2) ∑_{1}^{m} \frac{1}{|V |^{′4}} ⟨V, dΛ^{q}( ˘∇X^{i})(V)⟩^{′2} \\ & +\frac{1}{|V |^{′2}} ⟨V, −(R̆_{x}^{q})^{∗}(V)⟩′ . \end{aligned}  (5.0.6)

Then

H_{p}^{q}(V, V) = H̄_{p}^{q}(V, V) + 2 ∫_{0}^{t} \frac{⟨(dΛ^{q} ˘∇A)(V), V ⟩}{|V |^{′2}}′ds.  (5.0.7)

Let W_{s} = \frac{V_{s}}{|V_{s}|^{′2}} be the process projected on the unit sphere S(Λ^{q}T M) and set

M_{t} = p ∑_{i=1}^{m} ∫_{0}^{t} ⟨W_{s}, (dΛ^{q} ˘∇X^{i})(W_{s})⟩′ dB_{s}^{i}. Then

|V_{t}|^{′p} = |V_{0}|^{′p} exp (M_{t} − \frac{1}{2} ⟨M⟩_{t} + \frac{p}{2} ∫_{0}^{t} H_{p}^{q}(W_{s}, W_{s})ds) .  (5.0.8)

Let (x̃_{t}, Ṽ_{t}) be the solution and the derivative process of the stochastic differential equation

dx̃_{t} = X(x̃_{t}) ∘ dB̃_{t} + A(x̃_{t})dt

where

B̃_{t}^{i} = B_{t}^{i} − p ∫_{0}^{t} ⟨W_{s}, (dΛ^{q}) ˘∇X^{i}(−))(W_{s})⟩′ ds.

Then x_{·} and x̃_{·} have the same distribution from the defining property of ˘∇. By the Girsanov-Cameron-Martin theorem, if W̃_{s} = \frac{Ṽ_{s}}{|Ṽ_{s}|′},

𝔼|V_{t}|^{′p} = |V_{0}|^{′p} 𝔼 exp (\frac{p}{2} ∫_{0}^{t} H_{p}^{q}(W̃_{s}, W̃_{s})ds)

and so it follows that

𝔼 exp (\frac{1}{2} ∫_{0}^{t} \underline{h}_{p}^{q}(x_{s})ds) ≤ 𝔼[\frac{|V_{t}|^{′p}}{|V_{0}|^{′p}}] ≤ 𝔼 exp (\frac{1}{2} ∫_{0}^{t} h_{p}^{q}(x_{s})ds).

Remarks :

(i). The quantity ˘∇X^{i} which appears here and below is a generalization of the shape operator of a submanifold of ℝ^{m}. Indeed in the gradient system case

⟨ ˘∇_{v}X^{i}, u⟩_{x} = ⟨α(u, v), e_{i}⟩_{ℝ^{m}}

where α is the second fundamental form of the embedding determining X. See [Li94a],

[ER96] for geometric implications of Corollary 5.0.6 in the gradient case.

(ii). Fix x ∈ M and let ⟨, ⟩ be an inner product on T_{x}M extending that given by our s.d.e. on E_{x}. Let {e_{1}, …, e_{n}} be an orthonormal base for T_{x_{0}}M, ⟨, ⟩. As observed for the Levi-Civita connection in [ER96], see also [Ros97], the Weitzenbock curvatures acting on primitive vectors V = e_{1} ∧ … ∧ e_{q}, say, satisfy

⟨(R̆^{q})^{∗}V, V ⟩ = ∑_{j=1}^{q} ∑_{l≥q+1}^{λ} K̆(e_{j}, e′_{l})

where K̆ is the sectional curvature defined by

⟨R̆(v ∧ w), v ∧ w⟩ = K̆(v, w)
(= 0 unless v, w ∈ E_{x})

for R̆_{x} : ∧^{q}T M → ∧^{q}E_{x} the curvature operator and {e′_{1}, …, e′_{λ}} together with E_{x} ∩ {e_{1}, …, e_{q}} gives an orthonormal base for E_{x}.

Proof. First by Corollaries B.2 and C.5 of the Appendix,

∑_{r=1}^{m} ⟨δ^{2}Λ( ˘∇_{·}X^{r})(V), V ⟩

= −2 ∑_{r=1}^{m} ∑_{i<k,j<k} ⟨ ˘∇_{e_{j}}X^{r} ∧ ˘∇_{e_{k}}X^{r}, e_{i} ∧ e_{l}⟩ ⟨a_{j}a_{k}V, a_{l}a_{i}V ⟩

= 2 ∑ _{1≤j<k≤q}^{E} ⟨R̆_{x}(e_{j} ∧ e_{k}), e_{j} ∧ e_{k}⟩ where ∑^{E} refers to summation only over these j, k with e_{j}, e_{k} in E_{x}.

On the other hand

∑_{r=1}^{m} ⟨(dΛ^{q})R̆(X^{r}, −)X^{r}V, V ⟩ = ∑_{r=1}^{m} ∑_{j=1}^{q} ⟨R̆(X^{r}, e_{j})(X^{r}), e_{j}⟩ .

The result follows by choose {X^{r}(x)}_{r=1}^{dim(E)} to be the given orthonormal base for E_{x}.

The following corollary is immediate using [ER96] for the last part.

Corollary 5.0.6 Assume the stochastic differential equation has no explosion and Suppose that ˆ∇ is metric with respect to a metric ⟨, ⟩′ on T M. Then the stochastic dynamical system is strongly (q, p)-moment stable, with respect to the metric ⟨, ⟩′, if

lim sup_{t→∞} \frac{1}{t} sup_{x∈K} log 𝔼 exp (\frac{1}{2} ∫_{0}^{t} h_{p}^{q}(ξ_{s}(x))ds) < 0, all K compact.

It is not strongly (q, p)-moment stable if \underline{h}_{p}^{q} ≥ 0. In particular if

lim sup_{t→∞} sup_{x∈M} \frac{1}{t} log 𝔼 exp (\frac{1}{2} ∫_{0}^{t} h_{1}^{q}(ξ_{s}(x))ds) < 0

and M is compact then the homology group H_{q}(M, Z) vanishes.

Remarks: (i). In fact we have a more general formula than (5.0.5): Let ˜∇ be a semi-connection, metric with respect to ⟨, ⟩′. Write X^{0} ≡ A for simplicity and denote by S^{i}(t, x) the flow of the vector fields X^{i}, and T S^{i}(t, x)(v) the derivative flow. Then

\begin{aligned}|V_{T} |^{′p} = & |V_{0}|^{′p} + ∑_{1}^{m} p ∫_{0}^{T} |V_{s}|^{′p−2} ⟨V_{s}, \frac{D̃}{∂t} ∧^{q} T S^{i}(t, x_{s})(V_{s})|_{t=0}⟩′ dB_{s}^{i} \\ & +\frac{p}{2} ∫_{0}^{T} |V_{s}|^{′p}H̄_{p}^{q}(V_{s}, V_{s})ds, \end{aligned}  (5.0.9)

where

\begin{aligned}H̄_{p}^{q}(V, V) = & ∑_{i=1}^{m} \frac{1}{|V |^{′2}} |\frac{D̃}{∂t}Λ^{q}T S_{t}^{i}(V)|_{t=0}|^{′2} \\ & +(p − 2) ∑_{1}^{m} \frac{1}{|V |^{′4}} ⟨V, \frac{D̃}{∂t}Λ^{q}T S_{t}^{i}(V)|_{t=0}⟩^{′2} \\ & + ∑_{1}^{m} \frac{1}{|V |^{′2}} ⟨V, \frac{D̃^{2}}{∂t^{2}} Λ^{q}T S^{i}(t, x)(V)|_{t=0}⟩′ . \end{aligned}  (5.0.10)

This is deduced by an Itô’s formula in Elworthy [Elw88], see also Elworthy-Rosenberg [ER96] for gradient systems. Now take ˜∇ = ˆ∇ and observe that

\frac{D̂}{∂t}T S^{i}(t, x)(v) = ˆ∇X^{i}(S^{i}(t, x)) (T S^{i}(t, x)(v)) − T̆ (X^{i}(S^{i}(t, x)), T S^{i}(t, x)(v))
= ˘∇X^{i} (S^{i}(t, x)) (T S^{i}(t, x)(v)) ,

and similarly

\frac{D̂}{∂t}(Λ^{q}T S_{t}^{i}(t, x)(V)) = (dΛ^{q} ˘∇X^{i}) (Λ^{q}T S^{i}(t, x)(V)) .  (5.0.11)

Formula (5.0.5) now follows from Lemma 2.4.4, and (5.0.11).

(ii). If A ∉ E, let ∇^{1} be any extension of ˘∇ as in §1.3B and ∇^{1′} be its adjoint. Suppose ∇^{1′} is metric with respect to a metric ⟨, ⟩ ′on T M. Then the above result holds with H_{p}^{q} replaced by

H_{p}^{q}(V, V) = H̄_{p}^{q}(V, V) + 2 ∫_{0}^{t} \frac{⟨((dΛ^{q})∇^{1}A) (V), V ⟩}{|V |^{′2}}′ds.

C. From Theorem 3.3.8 and (3.3.17) we see that in the nonsingular case with M compact if ˆ∇ is adapted to some Riemannian metric on M then the conditional expectations 𝔼{T_{x_{0}}ξ_{t}| F^{x_{0}}} and 𝔼{T_{x_{0}}ξ_{t}^{−1}| F^{x_{0}}} are bounded processes in (t, ω) ∈ [0, T] × Ω, any T > 0. This boundedness can be with respect to any Riemannian metric on M, by compactness. In particular in the nondegenerate case these two processes conditioned on ξ_{T} (x_{0}) = y_{0}, will also be almost surely bounded on [0, T]×Ω, any y_{0} ∈ M. (As described in the proof of theorem below we can make sense of this for all, not just almost all y_{0}.) The following partial converse to this shows that the existence or not of a metric to which ˆ∇ is adapted is reflected rather drastically in the behaviour of the flow.

Theorem 5.0.7 Suppose our s.d.e (3.0.1) is nondegenerate and has no explosion. Fix x_{0} ∈ M and T > 0. Let {ξ_{t}(x_{0}) : 0 ≤ t < ∞} be the solution from x_{0}, with {T_{x_{0}}ξ_{t} : t ≥ 0} the derivative process at x_{0}. Suppose from y_{0} ∈ M the local conditional expectations

𝔼{T_{x_{0}}ξ_{t}| F^{x_{0}}} and 𝔼{T_{x_{0}}ξ_{t}^{−1}| F^{x_{0}}}, 0 ≤ t ≤ T,

when conditioned on ξ_{T} (x_{0}) = y_{0} are in L^{∞} uniformly in t ∈ [0, T], for some Riemannian metric on M for which R̆ic^{#} − ˘∇A is bounded. Then ˆ∇ is a metric connection for some metric on M.

Note: The conditioned process are the same in law as the conditional expectations, given the bridge process, of the processes T_{x_{0}}ξ_{t} and T_{x_{0}}ξ_{t}^{−1}, respectively, conditioned on ξ_{T} (x_{0}) = y_{0}. (Their expectations at time T are 𝔼{T_{x_{0}}ξ_{t}| ξ_{T} (x_{0}) = x_{0}} and 𝔼{T_{x_{0}}ξ_{t}^{−1}| ξ_{T} (x_{0}) = x_{0}}.)

Proof. By Theorem 3.3.7 and (3.3.17) the local conditional expectations are given by W_{t} and W_{t}^{−1} respectively. From [Car88], for example, the corresponding conditioned processes {W_{y_{0},t} : 0 ≤ t ≤ T }, {W_{y_{0},t}^{−1} : 0 ≤ t ≤ T } are given by the equations (3.3.11) and (3.3.17), for q = 0, but with x_{t} = ξ_{t}(x_{0}) replaced by the bridge, ξ_{y_{0},t}(x_{0}), i.e. ξ_{·}(x_{0}) conditioned to be at y_{0} at time T. From these equations they are seen to be mutually inverse. Let { ˆ//_{t} : 0 ≤ t ≤ T } be parallel translation along the bridge using ˆ∇. We have

\frac{d}{dt} (W_{y_{0},t}^{−1} ˆ//_{t}) = \frac{1}{2}W_{y_{0},t}^{−1} R̆ic^{#} ˆ//_{t} − W_{y_{0},t}^{−1} ˘∇A ˆ//_{t}

Whence W_{y_{0},t}^{−1} ˆ//_{t} is in L^{∞} uniformly in t ∈ [0, T] for the given metric. From this our assumption on W_{t} implies that { ˆ//_{t} : 0 ≤ t ≤ T } is in L^{∞}. The result follows from Theorem 1.3.8.

D. Finally we give the following Khasminski formula; c.f. [Car85], [Elw88],

Proposition 5.0.8 Suppose M is compact and our s.d.e. is nondegenerate and ˆ∇ is metric with respect to a metric ⟨, ⟩′. Let S^{q}(T M) be the unit sphere subbundle of ∧^{q}T M, ν an ergodic invariant measure for the process induced on S^{q}(T M), and λ^{n} ≤ … ≤ λ^{1} the corresponding Lyapunov exponents of the solution flow. Then for some choice of ν,

λ^{1} + … + λ^{q} = − ∫_{S(T M)} ∑_{1}^{m} ⟨V, (dΛ)^{q}( ˘∇X^{i})(V)⟩^{′2} dν

+\frac{1}{2} ∫_{S(T M)} ∑_{i=1}^{m} |(dΛ)^{q}( ˘∇X^{i})(V)|^{′2}dν

−\frac{1}{2} ∫_{S(T M)} ⟨V, (R̆^{q})^{∗}(V)⟩′ dν.

Proof. From [Car85], there exists an ergodic ν such that λ^{1}+…+λ^{q} = lim_{t→∞} \frac{1}{t} log |V_{t}(ω)| for ν ∗ P almost all (V_{0}, ω) in S^{q}(T M) × Ω. For such V_{0} we apply Itô’s formula to log |V_{t}|′ and use (5.0.3)

log |V_{t}|′ = log |V_{0}|′ + ∫_{0}^{t} \frac{d|V_{s}|′}{|V_{s}|′} − \frac{1}{2} ∫_{0}^{t} \frac{d|V_{s}|′d|V_{s}|′}{|V_{s}|^{′2}}

= log |V_{0}| + ∫_{0}^{t} \frac{(|V_{s}|′)^{−1} ∑_{1}^{m} ⟨V_{s,}(dΛ^{q} ˘∇X^{i})V_{s}⟩′ dB_{s}^{i}}{|V_{s}|′}

−\frac{1}{2} ∫_{0}^{t} \frac{(|V_{s}|′)^{−2} ∑_{1}^{m} ⟨V_{s,}(dΛ^{q} ˘∇X^{i})V_{s}⟩^{′2}}{|V_{s}|^{′2}} ds

+\frac{1}{2} ∫_{0}^{t} ∑_{i=1}^{m} \frac{1}{|V_{s}|^{′2}} |dΛ^{q}( ˘∇X^{i})(V_{s})|^{′2}ds

−\frac{1}{2} ∫_{0}^{t} ∑_{i=1}^{m} \frac{1}{|V_{s}|^{′4}} ⟨V, dΛ^{q}( ˘∇X^{i})(V)⟩^{′2} ds

+\frac{1}{2} ∫_{0}^{t} ∑_{i=1}^{m} \frac{1}{|V_{s}|^{′2}} ⟨V_{s}, −(R̆_{x}^{q})^{∗}(V_{s})⟩′ ds

= log |V_{0}| + ∫_{0}^{t} ∑_{1}^{m} ⟨W_{s,}(dΛ^{q} ˘∇X^{i})W_{s}⟩′ dB_{s}^{i}

− ∫_{0}^{t} ∑_{1}^{m} ⟨W_{s,}(dΛ^{q} ˘∇X^{i})W_{s}⟩^{′2} ds + \frac{1}{2} ∫_{0}^{t} ∑_{i=1}^{m} |dΛ^{q}( ˘∇X^{i})(W_{s})|^{′2}ds

+\frac{1}{2} ∫_{0}^{t} ∑_{i=1}^{m} ⟨W_{s}, −(R̆_{x}^{q})^{∗}(W_{s})⟩′ ds.

Taking the limit and by the standard ergodic theorem we have the result.

Proposition 5.0.9 Suppose X is injective, and ˆ∇ is adapted to some metric on M and A(x) ∈ E_{x} for each x with ˘∇A ≡ 0 (or more generally A is a Killing field for that metric). Then all the Lyapunov exponents vanish.

Proof. In the non-degenerate case with ˘∇A ≡ 0 this is immediate from Proposition 5.0.8 since all terms in the integrand there vanish. The general case comes from Corollary 1.3.6 which implies that the flows ξ_{t} will consist of isometries. Note that ˘∇A ≡ 0 implies that A(x) = ∑_{j=1}^{m} α_{j}X^{j}(x) some α_{j} ∈ ℝ, and hence that A is a Killing field if each X^{j} is.

Remarks (i). Under certain hypoellipticity condition on the stochastic differential equation and its derivative flow Baxendale showed in [Bax86] that vanishing of all the Lyapunov exponents implies that the flow consists of isometries for some Riemannian metric on M. He also showed that equality of the exponents holds if and only if there is a metric on M for which the flow consists of conformal transformations. See also [BS88].

(ii). We do not have examples satisfying the hypothesis of the proposition other than left or right invariant systems on Lie groups.

Appendices

A Universal Connections as L-W connection

A. Let π : E → M be a vector bundle over a manifold M with a surjective bundle map X from the trivial bundle π_{0} : M × ℝ^{m} → M, to π. The adjoint of the bundle map is denoted by Y : E^{∗} → M ×ℝ^{m}. We give E the induced metric so XY : E^{∗} → E is an isometry, and use the metric to identify E^{∗} with E, so π can be considered as a subbundle of π_{0} via Y. The fibre of E above x will be written E_{x}, and similarly for other bundles.

Let O(π) be the bundle of orthonormal frames of π with structure group O(p) for p =dim E_{x}, O(π_{0}) the principal bundle O(π_{0}) : M × O(m) → M with structure group O(m). The adapted frame bundle O(π_{0}, π) : O(M; M) → M is the bundle of adapted frames in O(π_{0}), i.e. O(M; M)_{x} consists of frames (x, u), u ∈ O(m) such that u (ℝ^{p} × 0) = Y (E_{x}).

[figure: M × O(m); O(M, M); h; O(m); M; M; id; O(p) × O(m − p); O(E); O(p); M; id]

B. Denote by ω_{0} the trivial connection form on O(π_{0}), i.e. ω_{0}(w, v) = T L_{a}^{−1}v = a^{−1}v for each (w, v) ∈ T_{(x,a)}(M × O(m)). The induced connection ω_{00} on O(π, π_{0}) is the restriction of ω_{0} to O(π, π_{0}) followed by the projection onto the Lie algebra o(p) × o(m − p). By Proposition 1.2 of volume II and Proposition 6.4 of volume I of Kobayashi and Nomizu ([KN69a],[KN69b]) we see that ω_{00} is a connection because ad(O(p) × O(m − p) sends the complement o(p, m − p) of o(p) + o(m − p) to itself.

Define

h : O(π_{0}, π) → O(π), by
u ↦ X(x) ∘ u |_{ℝ^{p}} .

Here we are using u for both the frame and its principal part. Let ω be the unique connection on O(π) satisfying (c.f. p.79 of Kobayashi-Nomizu [KN69a]):

1. It is the only connection such that the horizontal subspaces of ω_{00} are mapped into the horizontal subspaces of ω by h.

2. h^{∗}ω = h · ω_{00}, where h · ω_{00} is the o(p) component of ω_{00}.

This unique connection ω on O(π) (related to X) is the universal connection in the sense of Narisimhan and Ramanan when M is the Grassmann manifold and π the universal bundle, as is discussed later.

C. With no loss of generality we assume Y (E_{x}) = ℝ^{p} × 0. Let σ be a curve with σ(0) = x and σ̇(0) = v, with horizontal lifts in O(π_{0}), O(π_{0}, π) and O(π) respectively σ̃_{0}(t) = (σ(t), σ̃_{0}^{1}(t)), σ̃_{00}(t) = (σ(t), σ̃_{00}^{1}(t)), and σ̃(t) = (σ(t), σ̃^{1}(t)), for σ̃_{00}^{1}(0) = σ̃_{0}^{1}(0) = Id, and σ̃^{1}(0) = h (σ̃_{00}(0)). The covariant differentiation coming from ω will be denoted by ∇. We shall show this is in fact the L-W connection corresponding to X.

Recall that if ξ is a section of E and v ∈ T_{x}M, the L-W connection determined by X is given by

˘∇_{v}ξ(x) = X(x)d [Y (·)ξ(·)] (v)  (A.1)

and is the unique metric connection such that

if e ∈ [KerX(x)]^{⊥} , then (∇X^{e})_{x} = 0.

Here X^{e}(·) = X(·)(e).

Proposition A.1 The L-W connection ˘∇ induced by X is the unique connection ω on O(π) related to X as in §B.

Proof. Fix x ∈ M. Take e ∈Im(Y (x)). We only need to show that

(∇X^{e})_{x} = 0.

Here X(x)^{e} = X(x)(e). By definition,

(∇X^{e})_{x} (v) = σ̃(0)\frac{d}{dt} (σ̃(t))^{∗} X^{e} (σ(t))_{|t=0} .

But by the definition of the induced connection ω,

σ̃(t) = h (σ̃_{00}(t)) = X (σ(t)) (σ̃_{00}(t)|_{ℝ^{p}}) .

So

(σ̃(t))^{∗} = P_{ℝ^{p}} (σ̃_{00}(t))^{∗} Y (σ(t)) ,

where P_{ℝ^{p}} is the projection from ℝ^{m} to ℝ^{p}. Consequently

(∇X^{e})_{x} (v) = σ̃(0)\frac{d}{dt}P_{ℝ^{p}} (σ̃_{00}(t))^{∗} Y (σ(t)) X^{e} (σ(t))|_{t=0}
= σ̃(0)P_{ℝ^{p}} [(σ̃_{00}(0))^{∗}] \frac{d}{dt}Y (σ(t))X (σ(t)) (e)|_{t=0}

+σ(0)P_{ℝ^{p}} [\frac{d}{dt} (σ̃_{00}(t))^{∗}] |_{t=0}(e),

= σ(0)P_{ℝ^{p}} [\frac{d}{dt} (σ̃_{00}(t))^{∗}] |_{t=0}(e).

The last step comes from the fact that Y (x)X(x) is the projection P to Y (E_{x}) and P (0)Ṗ (0)P (0) = 0.

The required conclusion will now follow from the following lemma:

Lemma A.2

σ(0)P_{ℝ^{p}} [\frac{d}{dt} (σ̃_{00}(t))^{∗}] |_{t=0}Y (σ) = 0.

Proof. By skew adjointness,

\frac{d}{dt} [σ̃_{00}(t)]^{∗} |_{t=0} = −σ̇_{00}^{1}(0).

Since σ̃_{00}(t) is horizontal,

0 = ω_{00} ( ˙σ̃_{00}^{1}(t)) = o(p) × o(m − p) − component of ω_{0} ( ˙σ̃_{00}^{1}(t))
= o(p) × o(m − p) − component of (σ̃_{00}^{1}(t))^{−1} ˙ σ̃_{00}^{1}(t).

So (σ̃_{00}^{1}(t))^{−1} ˙ σ̃_{00}^{1}(t) belongs to o(p, m − p). In particular so does ˙ σ̃_{00}^{1}(0). The result follows.

D. Let M = G(p, q) be the Grassmann manifold of p-planes in ℝ^{m} for m = p + q, and π : B → M the vector bundle over M where the fibre B_{L} at L ∈ G(n, p) is the set of points in L. It is a Riemannian bundle. Let Y^{U} be the vector bundle map from π to the trivial bundle \underline{ℝ}^{m} defined by Y^{U}(L)(x) = i_{L}(x) for each x ∈ B_{L}. Here i_{L} is the inclusion of L onto ℝ^{m}. Its adjoint map X^{U} is given by X^{U}(L) = P_{L}, the orthogonal projection. Thus B is identified with a subbundle of \underline{ℝ}^{m}.

Let V (p, q) be the set of p-frames in ℝ^{m},i.e.

V (p, q) = {(e_{1}, …, e_{p}) |e_{i} ∈ ℝ^{m}, ⟨e_{i}, e_{j}⟩ = δ_{i,j} } .

This is the Stiefel manifold of p-frames in ℝ^{m}. With structure group O(p) it is the Stiefel bundle over G(p, q). It is in fact the associated principal bundle of B.

Consider the principal bundle π : O(m) → G(p, q) with structure group O(p) × O(q), where

π(T) ≡ T [ℝ^{p} × 0] ⊂ ℝ^{m}

for each T ∈ O(m). It turns out to be the bundle of adapted frames of π : V (p, q) → G(p, q) in π_{0} : G(p, q) × O(m) → G(p, q). This is because T ∈ O(m) is adapted if and only if T [ℝ^{p} × 0] = L = B_{L}. We are now in the picture of earlier discussions.

[figure: M × O(m); O(m); O(B); h; O(m); M; M; id; O(p) × O(q); O(p); M; id]

Now h : O(m) → V (p, q) is given by h(T) = T (ℝ^{n}). As before there is the connection ω on O(π).

Now V (p, q) → G(p, q) has a canonical connection ω_{U} as described by Narasimhan and Ramanan [NR61] as “the universal connection” via the connection form S^{T} dS, where S maps V (p, q) to m rows and p columns matrices in the following way: if α ∈ V (p, q) with α_{i} = ∑_{j=1}^{m} v_{i,j}e_{j}, where (e_{j}) is an orthonormal basis of ℝ^{m}, S(α) = (v_{j,i}). Denote by S(α)^{T} its transpose. Set

ω_{U} = S^{T} dS.

Then ω_{U} is a o(p)-valued 1-form (since S(α)^{T} S(α) = Id_{p×p}).

Lemma A.3 The L-W connection ω induced by X^{U} : \underline{ℝ}^{m} → B is given by the universal connection S^{T} dS on the Stiefel bundle.

Proof. We only need to show that the connection ω is given by S^{T} dS, i.e. that h^{∗}(S^{T} dS) is the o(p)-component of ω_{00}. This is clear since for A ∈ o(m), h^{∗}(S^{T} dS)(A) = (S∘)^{T} d(S ∘ h)(A) = P_{ℝ^{p}}Ai_{ℝ^{p}}.

Given our surjection X : \underline{ℝ}^{m} → E with adjoint Y so that Y (x) : E_{x} → ℝ^{m}, there is a bundle homomorphism Φ from O(E) to the Stiefel bundle,

[figure: Φ:; O(E); V (p, q); M; G(p, q); Φ_{0}]

defined by:

Φ_{0}(x) = Image Y (x),
Φ(u) = (Y (x)u(e_{1}), …, Y (x)u(e_{p})) ,

where (e_{i}) is an orthonormal basis for ℝ^{p}.

Conversely any such bundle homomorphism Φ : O(E) → V (p, q) over Φ_{0} : M → G(p, q) comes from a surjective map: indeed for x ∈ M, take a frame u ∈ O(E) at x, and set

Y (x) = Φ(u) ∘ u^{−1} : T_{x}M → ℝ^{m}.

This is independent of the choice of u. Here we have used Φ(u) for the induced transformation: ℝ^{p} → ℝ^{m}. Set X(x) = Y (x)^{∗}.

On the vector bundle level let E^{1} → M_{1} and E^{2} → M_{2} be two vector bundles over manifolds and let f be a vector bundle map which is a 1-1 map on fibres over f_{0} : M_{1} → M_{2}. Given a vector bundle map X_{2} : \underline{ℝ}^{m} → E^{2} there is an induced map X_{1} : \underline{ℝ}^{m} → E^{1} given by

X_{1}(x)(e) = (f)_{f_{0}(x)}^{−1} X_{2} (f_{0}(x)) (e).

[figure: E^{1}; E^{2}; f; M_{1}; M_{2}; f_{0}; ^{\underline{ℝ}^{m}}; X_{2}]

This gives E^{1} an induced metric and makes f an isometry on fibres. Let Γ^{X_{2}} be the L-W connection on O(E^{2}) from X_{2} and Γ^{X_{1}} be the L-W connection on O(E^{1}) from X_{1}. Let f^{∗} (Γ^{X_{2}}) be the pull back of Γ^{X_{2}}.

Lemma A.4

f^{∗} (Γ^{X_{2}}) = (Γ^{X_{1}}) .

Proof. Let Y_{1} and Y_{2} be respectively the adjoint of X_{1} and X_{2}. Take e ∈ Image(Y_{1}(x)), say e = Y_{1}(v_{e}). We need only to show

(∇X_{1}^{e})_{x} = 0.

Here ∇ is the covariant differentiation corresponding to the pulled back connection. Take a curve σ_{1} ∈ M_{1} with σ_{1}(0) = x. Let σ̃_{1} be the horizontal lift of σ_{1}(·) in the frame bundle of E^{1}, and σ̃_{2} the horizontal lift of f_{0} (σ_{1}(·)). Then

f (σ̃_{1}(t)) = σ̃_{2}(t).

By definition,

(∇X_{1}^{e})_{x} = σ̃_{1}(0)\frac{d}{dt} (σ̃_{∗}(t))^{−1} X_{1}^{e} (σ(t)) |_{t=0}
= f^{−1} (σ̃_{2}(0)) \frac{d}{dt} (σ̃_{2}(t))^{−1} X_{2}^{e} (f_{0}(σ(t))) |_{t=0}

The right hand side is zero from the characterization of L-W connections and the observation that e is in the image of Y_{2} since e = Y_{1}(x)(v_{e}) = Y_{2}(f_{0}(x)) (f(v_{e})).

Theorem A.5 Every metric connection on E is a L-W connection for some X : \underline{ℝ}^{m} → E, some finite m.

Proof. By Narasimhan and Ramanan, any metric connection form ω_{g} is given by ω_{g} = Φ^{∗}(ω_{U}) from some bundle homomorphism Φ : O(E) → V (p, q). Define X, Y from Φ as before.

[figure: E; B; f; M; G(p, q); ^{\underline{ℝ}^{m}}; X_{U}]

Set f = (X^{U}Y). Then Φ is induced by f and so, by lemma A.3, Φ^{∗}(ω_{U}) is the connection form for f^{∗}(Γ^{X^{U}}). By Lemma A.4 f^{∗} (Γ^{X^{U}} ) = Γ^{X}. So ω_{g} is the connection form for the L-W connection Γ^{X}.

B Creation and Annihilation operators (notation for section 2.4)

Let A : V → V be a linear map on an inner product space V and (e_{1}, …, e_{n}) an orthonormal base for V. There are the operators (dΛ)^{∗}(A) on the space of tensor products of ∧^{∗}V, which restricts to ∧^{p}V to give (dΛ)^{p}(A):

(dΛ)^{p}(A) (u_{1} ∧ … ∧ u_{p}) = ∑_{1}^{p} u_{1} ∧ … ∧ u_{j−1} ∧ Au_{j} ∧ u_{j+1} ∧ … ∧ u_{p},

and (δ^{2}Λ)^{∗} defined by:

(δ^{2}Λ)^{∗}A = (dΛ)^{∗}A ∘ (dΛ)^{∗}A − (dΛ)^{∗}A^{2}.

If ϕ is a linear map on ∧^{p}V, we define Âϕ(v) = ϕ(Av) and so the last linear transformations we defined on ∧^{p}V give:

ˆ(dΛ)^{p}(A)(ϕ)(u_{1}, …, u_{p}) = ∑_{j=1}^{p} ϕ(u_{1}, …, Au_{j}, …, u_{p}),

and for p > 1

(δ̂^{2}Λ^{∗})A(ϕ)(u_{1}, …, u_{p}) = ∑_{i≠j} ϕ(u_{1}, …, Au_{i}, …, Au_{j}, …, u_{p}).

And so

(δ̂^{2}Λ^{∗})A(ϕ) = (d̂Λ^{∗}) ∘ (d̂Λ^{∗})A(ϕ) − (d̂Λ^{∗})A^{2}(ϕ),

Let a_{j}^{∗} be the ”creation operator”on ∧^{∗}V, a_{j}^{∗}v = e_{j} ∧v if (e_{1}, …, e_{n}) is an orthonormal basis for ∧^{∗}V, and a_{j} its adjoint, the ”annihilation operator”. For linear forms we have the corresponding operators: (a^{j})^{∗}ϕ(v) = ϕ(a_{j}v) and (a^{j}ϕ)(v) = ϕ(a_{j}^{∗}v). In particular a^{j}ϕ(v) = ϕ(e_{j} ∧ v) and (a^{j})^{∗}ϕ(v) = e_{j}^{∗} ∧ ϕ .

Then

dΛ^{∗}(A) = ∑_{i,j} ⟨Ae_{j}, e_{i}⟩ a_{i}^{∗}a_{j}.  (B.1)

See [CFKS87].

Lemma B.1

(δ^{2}Λ)^{∗}A = − ∑_{i,j,k,l} ⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{l}⟩ a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}.

Proof. By (B.1),

(dΛ)^{∗} ∘ (dΛ)^{∗}(A)(−) = ∑_{i,j} ⟨Ae_{j}, e_{i}⟩ a_{i}^{∗}a_{j} ((dΛ)^{∗}(A)(−))

= ∑_{i,j,k,l} ⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{l}⟩ a_{i}^{∗}a_{j}a_{l}^{∗}a_{k}(−)

= − ∑_{i,j,k,l} ⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{l}⟩ a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}(−)

+ ∑_{i,j,k} ⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{j}⟩ a_{i}^{∗}a_{k}(−)

The last step comes from the identity a_{j}a_{l}^{∗} = −a_{l}^{∗}a_{j} + δ_{jl}, as used in [CFKS87]. However

∑_{i,j,k} ⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{j}⟩ a_{i}^{∗}a_{k}(−) = ∑_{i,j,k} ⟨A^{2}e_{k}, e_{i}⟩ a_{i}^{∗}a_{k}(−) = (dΛ^{∗})(A^{2}).

Consequently,

(dΛ)^{∗} ∘ (dΛ)^{∗}(A)(−)

= − ∑_{i,j,k,l} ⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{l}⟩ a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}(−) + (dΛ)^{∗}(A^{2})(−),

Corollary B.2 The map (δ^{2}Λ)^{∗}A : V → V can be written as:

(δ^{2}Λ)^{∗}A = −2 ∑_{i<l,j<k} ⟨Ae_{j} ∧ Ae_{k}, e_{i} ∧ e_{l}⟩ a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}.

Proof. Note a_{j}^{∗}a_{j}^{∗} = 0. We split the sum in (δ^{2}Λ)^{∗}A into two parts: i < l and i > l. After rearrangements, we have:

(δ^{2}Λ)^{∗}(A)

= − ∑_{i<l,1≤k,j≤m} [⟨Ae_{j}, e_{i}⟩ ⟨Ae_{k}, e_{l}⟩ − ⟨Ae_{j}, e_{l}⟩ ⟨Ae_{k}, e_{i}⟩] a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}

= − ∑_{i<l,1≤k,j≤m} ⟨Ae_{j} ∧ Ae_{k}, e_{i} ∧ e_{l}⟩ a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}

= −2 ∑_{i<l,k<j} ⟨Ae_{j} ∧ Ae_{k}, e_{i} ∧ e_{l}⟩ a_{i}^{∗}a_{l}^{∗}a_{j}a_{k}.

There is a similar expression for linear forms from

(d̂Λ^{∗}A)(ϕ) = ∑_{i,j} ϕ(⟨Ae_{i}, e_{j}⟩ a_{i}^{∗}a_{j}) = ∑_{i,j} ⟨Ae_{j}, e_{i}⟩ (a^{i})^{∗}a^{j}ϕ :

Corollary B.3 The map (δ̂^{2}Λ^{∗})A : ∧^{∗}V^{∗} → ∧^{p}V^{∗} is given by:

\binom{}{δ^{2}Λ}^{∗} A(ϕ) = − ∑_{i,j,k,l} ⟨Ae_{i}, e_{j}⟩ ⟨Ae_{k}, e_{l}⟩ (a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l} ϕ
= −2 ∑_{i<k,j<l} ⟨Ae_{i} ∧ Ae_{k}, e_{j} ∧ e_{l}⟩ (a^{i})^{∗}(a^{k})^{∗}a^{j}a^{l} ϕ.

C Basic formulae

In this section we give some basic formulae for ˘∇ and T̆ in terms of the defining map X or the Gaussian field W.

Let W be a mean zero Gaussian field of sections Γ(E) of a vector bundle E, ˘∇ the associated L-W connection, and let ⟨, ⟩_{x} be the metric on E induced from the Gaussian structure, as given in section 1.1C. Recall (1.1.5):

˘∇_{v}U = 𝔼 W (x) \frac{d}{dt} ⟨U(σ(t)), W (σ(t))⟩_{σ(t)}|_{t=0} v ∈ T_{x}M, U ∈ ΓE,  (C.1)

where σ is any C^{1} smooth curve with σ̇(0) = v, and also recall the expansion U = 𝔼W ⟨u, W ⟩ for U ∈ Γ(E).

Proposition C.1 1. For any connection ˜∇ on E,

˜∇_{v}U = 𝔼 ˜∇_{v}W ⟨U, W ⟩_{x_{0}} + ˘∇_{v}U.  (C.2)

2. 𝔼 ˘∇_{W} W = 0.

3. A connection ˜∇ on E is adapted to ⟨, ⟩_{x} if and only for any U ∈ Γ(E) and tangent vector v,

𝔼 ˜∇_{v}W ⟨U, W ⟩ + 𝔼 W ⟨U, ˜∇_{v}W ⟩ = 0.  (C.3)

Proof. Take v ∈ T_{x_{0}}M and let σ be a C^{1} smooth curve with σ̇(0) = v. Expanding U (σ(t)) in W we see:

˜∇_{v}U = \frac{D̃}{dt}𝔼 W (σ(t)) ⟨U(σ(t)), W (σ(t))⟩_{σ(t)}|_{t=0}

= 𝔼 ˜∇_{v}W ⟨U, W ⟩_{x_{0}} + 𝔼 W (x_{0}) \frac{d}{dt} ⟨U(σ(t)), W (σ(t))⟩|_{t=0}
= 𝔼 ˜∇_{v}W ⟨U, W ⟩_{x_{0}} + ˘∇_{v}U.

Thus (C.2). Take ˜∇ = ˘∇ in (C.2)) to see part (2) (or use the defining property of Proposition 1.1.3).

Next suppose that ˜∇ is adapted to the metric. Applying ˜∇_{v} in the second line of the earlier calculation, we have

˜∇_{v}U = 𝔼 ˜∇_{v}W ⟨U, W ⟩ + 𝔼 W ⟨U, ˜∇_{v}W ⟩ + ˜∇_{v}U

and thus (C.3).

On the other hand (C.3) implies

𝔼 ⟨ ˜∇_{v}W, U⟩ ⟨U, W ⟩ = 0.

Consequently,

d ⟨U, U⟩ (v) = 2 ⟨ ˘∇_{v}U, U⟩ (v)
= 2 ⟨ ˜∇_{v}U, U⟩ − 2𝔼 ⟨ ˜∇_{v}W, U⟩ ⟨U, W ⟩
= 2 ⟨ ˜∇_{v}U, U⟩ ,

i.e. ˜∇ is metric.

Corollary C.2 Let ˜∇ be a connection on a subbundle E of T M. Denote by T̆ and T̃ respectively the torsions for ˘∇ and ˜∇. Then for sections U, Z of E with U(x_{0}) = u and Z(x_{0}) = z,

T̆(z, u) − T̃ (z, u) = 𝔼 W (x_{0}) ⟨u, ˜∇_{z}W ⟩ − 𝔼 W (x_{0}) ⟨z, ˜∇_{u}W ⟩  (C.4)
= −𝔼 ⟨W (x_{0}), u⟩ ˜∇_{z}W + 𝔼 ⟨W (x_{0}), z⟩ ˜∇_{u}W  (C.5)

and

T̆(z, u) = [Z, U](x_{0}) + 𝔼[W, Z](x_{0}) ⟨W, u⟩_{x_{0}} − 𝔼[W, U](x_{0}) ⟨W, z⟩_{x_{0}} .  (C.6)

Proof. The first formula follows from (1.3.12) and the second from Proposition C.1 and (C.5).

Finally we write ˘∇ in terms of the vector fields Z^{·} as defined by (1.1.1):

Proposition C.3 Let ˜∇ be a connection on a subbundle Ẽ of T M containing E. Then for U ∈ Γ(E) and V ∈ Γ(T M) with U(x_{0}) = u and V (x_{0}) = v,

˘∇_{v}U = ˜∇_{v}U − ˜∇_{v}Z^{u},  (C.7)
ˆ∇_{u}V = ˜∇′_{u}V − ˜∇_{v}Z^{u}.  (C.8)

Proof. The first identity is just (1.1.3). The second follows from (1.3.1) and (C.7):

ˆ∇_{u}V = ˘∇_{v}U + [U, V](x_{0}) = ˜∇_{v}U − ˜∇_{v}Z^{u} + [U, V](x_{0}) = ˜∇′_{u}V − ˜∇_{v}Z^{u}.

Formulae related to curvatures

First we give a formula for the curvature tensor R̆ in terms of X or W. From this a series of identities and inequalities relating the quantities of ˘∇ with those of ˆ∇ are observed. In particular we give an interpretation of the form T̆^{#}, derived from torsion T̆, being closed or co-closed.

A. Let E be a subbundle of T M, R̆ : T M × T M → L(E; E) the curvature tensor for a metric connection ˘∇, and W a Gaussian vector field which induces ˘∇ in the sense of section 1.1.

Proposition C.4 For u ∈ E_{x_{0}}, and v_{1}, v_{2} ∈ T_{x_{0}}M,

R̆(v_{1}, v_{2})(u) = 𝔼 ˘∇_{v_{1}}W ⟨ ˘∇_{v_{2}}W, u⟩ − 𝔼 ˘∇_{v_{2}}W ⟨ ˘∇_{v_{1}}W, u⟩ .  (C.9)

In the metric form,

R̆(v_{1}, v_{2})(u) = ∑_{i} ˘∇_{v_{1}}X^{i} ⟨ ˘∇_{v_{2}}X^{i}, u⟩ − ∑_{i} ˘∇_{v_{2}}X^{i} ⟨ ˘∇_{v_{1}}X^{i}, u⟩ .  (C.10)

Proof. First recall formula (1.3.12) from section 1.3:

˘∇_{v}U = [V, U] + 𝔼[W, V] ⟨W, U⟩  (C.11)

Let U be a horizontal vector field and V, Z vector fields,then

˘∇_{Z}[ ˘∇_{V} U] = [Z, ˘∇_{V} U] + 𝔼 [W, Z] ⟨W, ˘∇_{V} U⟩
= [Z, [V, U]] + 𝔼 [Z, [W, V]] ⟨W, U⟩

+𝔼 [W, V] d ⟨W, U⟩ (Z(·)) + 𝔼[W, Z] ⟨W, ˘∇_{V} U⟩ .

Use Jacobi’s identity twice to obtain

R̆(Z, V)U := ˘∇_{Z} ˘∇_{V} U − ˘∇_{V} ˘∇_{Z}U − ˘∇_{[Z,V]}U
= 𝔼 { [W, V] d ⟨W, U⟩ (Z(·))
− [W, Z] d ⟨W, U⟩ (V (·)) }

+𝔼 {[W, Z] ⟨W, ˘∇_{V} U⟩ − [W, V] ⟨W, ˘∇_{Z}U⟩} .

Take U = Z^{u} and V, Z with V (x_{0}) = v, Z(x_{0}) = z for z, v ∈ T_{x_{0}}M, and u ∈ E_{x_{0}}. Then

R̆(z, v)u = 𝔼 {[W, V] ⟨ ˘∇_{z}W, u⟩ − [W, Z] ⟨ ˘∇_{v}W, u⟩}
= 𝔼 { ˘∇_{z}W ⟨ ˘∇_{v}W, u⟩ − ˘∇_{v}W ⟨ ˘∇_{z}W, u⟩} ,

using (1.3.1) and the independence of W (x) and ˘∇_{·}W |_{T_{x_{0}}M}.

Consequently for v_{i} ∈ T_{x}M, u_{i} ∈ E_{x},

⟨R̆(v_{1}, v_{2})u_{1}, u_{2}⟩ = −𝔼 ⟨ ˘∇_{v_{1}}W ∧ ˘∇_{v_{2}}W, u_{1} ∧ u_{2}⟩_{Λ^{2}E_{x}} .

Corollary C.5 Let R̆ : Λ^{2}T M → Λ^{2}E be the curvature operator defined by

⟨R̆(v_{1} ∧ v_{2}), u_{1} ∧ u_{2})⟩ = ⟨R̆(v_{1}, v_{2})u_{2}, u_{1}⟩.

Then

R̆ = 𝔼 ˘∇_{·}W ∧ ˘∇_{·}W.

In the metric form,

R̆ = ∑_{i=1}^{m} ˘∇_{·}X^{i} ∧ ˘∇_{·}X^{i}.

Remark: For A : T M ⊕ KerX → E the ’shape operator’, defined by

A(u, e) = ˘∇X(e)(u),

the proposition gives

R̆(u, v)w = trace {A(u, −) ⟨A(v, −), w⟩ − A(v, −) ⟨A(u, −), w⟩}

which reduces in the gradient case to Gauss’s equation for the curvature of a submanifold in ℝ^{m} (e.g. p. 23 [KN69b]).

B. Let ˜∇ and ∇^{0} be two connections on E with D̃ : T M × E → E the bilinear map defined by :

˜∇_{V} U = ∇_{V}^{0} U + D̃(V, U).  (C.12)

Their curvature tensors are related by the following formula: for v_{1}, v_{2} and u in E_{x},

\begin{aligned}R̃(v_{1}, v_{2})u & = R^{0}(v_{1}, v_{2})u + (∇_{v_{1}}^{0}D̃) (v_{2}, u) − (∇_{v_{2}}^{0}D̃) (v_{1}, u) \\ & +D̃(v_{1}, D̃(v_{2}, u)) − D̃(v_{2}, D̃(v_{1}, u)) + D̃ (T^{0}(v_{1}, v_{2}), u) , \end{aligned}  (C.13)

and the two covariant derivatives of D̃ restricted to E × E are related by

( ˜∇_{v_{1}} D̃) (v_{2}, u) − (∇_{v_{1}}^{0}D̃) (v_{2}, u)
= D̃ (v_{1}, D̃(v_{2}, u)) − D̃ (v_{2}, D̃(v_{1}, u)) − D̃ (D̃(v_{1}, v_{2}), u) , v_{1}, v_{2}, u ∈ E_{x_{0}}

In the nondegenerate case, we shall take ∇^{0} = ∇, the Levi-Civita connection for the induced metric. Recall the differential 3-form T^{#} defined in section 2.2.

Proposition C.6 Let ˘∇ be a torsion skew symmetric connection on T M. If R, R̆ and R̂ denote respectively the curvature tensors for ∇, ˘∇ and ˆ∇ then:

\begin{aligned}R̆(v_{1}, v_{2})u = & R(v_{1}, v_{2})u + \frac{1}{2} (∇_{v_{1}} T̆) (v_{2}, u) − \frac{1}{2} (∇_{v_{2}} T̆) (v_{1}, u) \\ & +\frac{1}{4} T̆(v_{1}, T̆(v_{2}, u)) − \frac{1}{4} T̆(v_{2}, T̆(v_{1}, u)). \end{aligned}  (C.14)

In particular,

1.

R̆ic(u, v) = Ric(u, v) − \frac{1}{2}δT̆^{#}(u, v) − \frac{1}{4}tr ⟨T̆(−, u), T̆(−, v)⟩ ,  (C.15)

so R̆ic(−, −) is symmetric if and only if δT̆^{#} = 0.

2.

R̆ic(u, v) − R̂ic(u, v) = −δT̆^{#}(u, v).  (C.16)

3.

R̆(v_{1}, v_{2})u = R̂(v_{1}, v_{2})u − ( ˘∇_{v_{1}} T̆) (v_{2}, u) + ( ˘∇_{v_{2}} T̆) (v_{1}, u).  (C.17)

⟨R̆(v_{1}, v_{2})u, w⟩ − ⟨R̂(u, w)v_{1}, v_{2}⟩ = \frac{1}{2}dT̆^{#}(v_{1}, v_{2}, u, w)  (C.18)

and

R̆ic(u, v) = R̂ic(v, u),  (C.19)

Remark: See also Lemma 3.5 in [Dri97a] for symmetricity of R̆ic in the torsion skew symmetric case.

Proof. First note that ˘∇ = ∇+ \frac{1}{2} T̆ in the torsion skew symmetric case. Equation (C.14) follows straightaway from (C.13). Let X : ℝ^{m} → T M be a map which has ˘∇ as its L-W connection. Recall R̆ic(u, v) = ∑ ⟨R̆(X^{i}, u)v, X^{i}⟩ so that

R̆ic(u, v) − Ric(u, v) = \frac{1}{2} ⟨∇_{X^{i}} T̆(u, v), X^{i}⟩ − \frac{1}{2} ⟨(∇_{u}T̆)(X^{i}, v), X^{i}⟩

−\frac{1}{4} ⟨T̆(u, T̆(X^{i}, v), X^{i}⟩ .

The first term of the right hand side is \frac{1}{2}δT̆^{#}. On the other hand the torsion skew symmetry gives

( ˘∇_{z}T̆) (u, v) = (∇_{z}T̆) (u, v) + Cyl T̆ (z, T̆(u, v))  (C.20)

and thus

(∇_{u}T̆) (v, v) = ( ˘∇_{u}T̆) (v, v) = 0.

It follows that the second term −\frac{1}{2} ⟨(∇_{u}T̆)(X^{i}, X^{i}), v⟩ vanishes. The last terms is now

−\frac{1}{4} ⟨T̆(X^{i}, T̆(u, X^{i})), v⟩ = \frac{1}{4} ⟨T̆(X^{i}, v), T̆(u, X^{i})⟩
= −\frac{1}{4}tr ⟨T̆(−, u), T̆(−, v)⟩ .

We have proved (C.15). Apply (C.15), and (C.14) respectively to both ˘∇ and ˆ∇ to obtain (C.16) and (C.17). Equation (C.18) follows from

⟨( ˘∇_{u}T̆)(w, v_{1}), v_{2}⟩ = ⟨( ˘∇_{u}T̆)(v_{1}, v_{2}), w⟩ ,

and (C.19) from (C.18).

Corollary C.7 If ˘∇ is a torsion skew symmetric connection on T M,

R̆ic(u, u) ≤ Ric(u, u),  (C.21)

Furthermore if k̆, k̂, and k are respectively the scaler curvature of the connections ˘∇, ˆ∇ and ∇. Then

k̆ = k̂ = k − \frac{1}{4}|T̆(−, −)|^{2}.

Remarks:

(i) When M is a Lie group with ˘∇ the left invariant connection and ⟨, ⟩ is biinvariant then R̆ ≡ 0, ∇T̆ ≡ 0 and (C.14) reduces to the standard formula for the curvature

R(v_{1}, v_{2})u = −\frac{1}{4} [ [Z^{v_{1}}, Z^{v_{2}}], Z^{u}]

by (1.3.6) and the Jacobi identity.

In this case (C.18) shows that T̆^{#} is a closed form, as is well known: for non-Abelian compact Lie groups it represents a non-trivial class in H^{3}(G), [Car36], which is clear since (C.16) shows it is harmonic. Indeed we obtain the following:

if M is compact with dimM ≥ 3 and admits a torsion skew symmetric connection with nonzero torsion which is flat together with its adjoint connection then H^{3}(M; ℝ) ≠ 0.

This would be an extension of Cartan’s result if the existence of such a connection were known on any manifolds other than Lie groups.

(ii) The inequality (C.21) does not hold in general without the assumption of torsion skew symmetry, even when ˆ∇ is adapted to some metric. A class of counter examples is provided by Lie groups with left invariant metrics having negative curvature, e.g. see [Mil76].

D List of notation

M  basic manifold,
H  Hilbert space, =ℝ^{m} if finite dimensional
X(x)  bundle homomorphism between the trivial bundle M × H and E over M,
Y (x) = X(x)^{∗} : T_{x}M → H,
N(x) = KerX(x) (often aasume of constant rank);
Z^{v} = X(·)Y (x_{0})v, v ∈ T_{x_{0}}M
⟨, ⟩ or ⟨, ⟩^{X}  metric induced on E from X,
⟨, ⟩^{1}  a Riemannian metric on T M = E ⊕ E^{⊥}, extending the metric induced

on E by X, having E^{⊥} orthogonal to E

⟨, ⟩′  a Riemannian metric not necessarily coming from X;
˘∇,  the connection associated to X, R̆, R̆ic, R̆^{q} its curvature, Ricci

curvature and Weitzenbock terms, and T̆ its torsion tensor

ˆ∇  the adjoint semi-connection of ˘∇, R̂, R̂ic, R̂^{q}, its curvature, Ricci

curvature, Weitzenbock terms and T̂ its torsion tensor,

∇  Levi-Civita connection, R, Ric, R^{q}, its curvature, Ricci curvature,

and Weitzenbock terms

∇^{1} = ˘∇ ⊕ ∇^{⊥}, the direct sum connection of ˘∇ on E and a connection

∇^{⊥} on E^{⊥}, T^{1} its torsion

˜∇  any connection, R̃ its curvature tensor, R̃ic its Ricci curvature, R̃^{q}

its Weitzenbock terms, ˜//_{·} the corresponding parallel translation, T̃ its torsion tensor, R̃_{ijkl} the associated curvatures,

R̃ic^{#}(v) = ∑_{1}^{m} R̃ic(v, X^{i}(x))X^{i}(x),

T̃^{#}  the 3-form related to T̃,
W  Gaussian field of sections of E,
D  (2,0)-tensor, the difference between two linear connections,
A, A^{q}  infinitesimal generator associated with a given sde and its

restriction on q-forms,

P_{t}  semigroup associated with stochastic flows;
A^{X} = \frac{1}{2} ∑_{j}^{m} ∇X^{j}(X^{j}) + A, where A the drift coefficient of the s.d.e. involved,
δ̂  ’divergence’ operator associated to ˘∇,
a^{i}, (a^{i})^{∗}  annihilation and creation operator,
ξ_{t}(x_{0}), ρ(x_{0})  solution to sde with initial point x_{0} and life time ρ(x_{0}),
x_{t} = ξ_{t}(x_{0}),

T ξ_{t}  the derivative process associated to ξ_{t},
v_{t}  often T_{x_{0}}ξ_{t}(v_{0}); for v_{0} ∈ T_{x_{0}}M,
W̆_{t}^{q,A}  the solution to covariant equation (3.3.11) involving R̆^{q}
β_{t}, B̃_{t}  orthogonal decomposition of the Brownian motion B_{t}

on ℝ^{m} using ˘∇ ˜//_{t}^{−1}dB_{t} = dβ_{t} + B̃_{t},

B̆_{t} = X(x_{0})B̃_{t}, the martingale part of the stochastic anti-development of

ξ_{t}(x_{0})

d_{E}f  the restriction of df to E
DiffM  C^{∞} diffeomorphisms of M.

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