RADEMACHER’S THEOREM ON CONFIGURATION SPACES AND APPLICATIONS

Michael Röckner and Alexander Schied

Abstract: We consider an L^{2}-Wasserstein type distance ρ on the configuration space Γ_{X} over a Riemannian manifold X, and we prove that ρ-Lipschitz functions are contained in a Dirichlet space associated with a measure on Γ_{X} satisfying some general assumptions. These assumptions are in particular fulfilled by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of ρ for a set to be exceptional. This result immediately implies, for instance, a quasi-sure version of the spatial ergodic theorem. We also show that ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form.

0. Introduction.

Let Γ_{X} be the configuration space over a Riemannian manifold X. In this paper, we consider a class of probability measures on Γ_{X}, which in particular contains certain Ruelle type Gibbs measures and mixed Poisson measures. Using a natural ‘non-flat’ geometric structure of Γ_{X}, recently analyzed in Albeverio, Kondratiev and Röckner (1996a), (1996b), (1997a), and (1997b), one can define weak derivatives and introduce the related Sobolev spaces. Here we are interested in a more detailed description of this concept of differentiability. Similar to the case of H-differentiability on Wiener space, it turns out that not all values which a function u takes in a small neighborhood of some γ ∈ Γ_{X} are relevant for its weak gradient ∇^{Γ}u(γ), but only those which are located in certain ‘directions’. Here, of course, the word ‘direction’ needs to be defined because of the absence of any linear structure on Γ_{X} even if X = ℝ^{d}.

Research supported in part by Deutsche Forschungsgemeinschaft and, at MSRI, by NSF grant DMS-9701755.

Our way of making the above precise is to prove an infinite dimensional version of the celebrated theorem of Rademacher (1919) stating that Lipschitz functions on ℝ^{d} are differentiable almost everywhere and and in the weak sense. On abstract Wiener space and its generalizations, similar results were obtained by Kusuoka (1982a), (1982b), Enchev and Stroock (1993), and Bogachev and Mayer-Wolf (1996). On configuration space, the correct Lipschitz condition is defined through an L^{2}-Wasserstein type distance function ρ, which, for non-compact X, divides Γ_{X} into uncountably many disjoint ‘fibers’, each of the form {ω | ρ(γ, ω) < ∞}. A consequence of our Rademacher type theorem then is that only the behavior of u in small ρ-balls around γ matters for the value of ∇^{Γ}u(γ).

In a second result, we prove a partial converse to our Rademacher theorem. It also allows us to identify ρ as the intrinsic metric of certain Dirichlet forms associated with our weak gradient. The resulting variational formula can also be regarded as a Kantorovich-Rubinstein type theorem for our L^{2}-Wasserstein metric. On abstract Wiener space such a converse to Rademacher’s theorem was obtained in Enchev and Stroock (1993) by a different method. As a by-product to our proof we obtain that all measures satisfying our assumptions have full topological support on Γ_{X}, a result which might be of independent interest, in particular for the Ruelle measures mentioned above.

Another main part of this paper is devoted to applications of the above results to the potential theory on configuration space. In particular, we show that if A ⊂ Γ_{X} has full measure, then the set of all points with positive ρ-distance to A is exceptional. This, for instance, implies immediately a quasi-sure version of the spatial ergodic theorem on Γ_{ℝ^{d}}. We also give a short proof of the quasi-regularity of our Dirichlet forms.

The organization of the paper is as follows. In Section 1, we describe our setup and the Rademacher type results under some general conditions on a measure μ on Γ_{X}. In Section 2, we identify a class of Ruelle type Gibbs measures which satisfy these assumptions. The applications to potential and ergodic theory are presented in Section 3. Sections 4 and 5 are devoted to a detailed analysis of the topological and geometric properties of our L^{2}-Wasserstein distance ρ. These results serve as a preparation for the proofs of our main theorems, but some may also be of independent interest. Proofs concerning our various Sobolev spaces and Dirichlet forms are given in Section 6. Finally the proofs of our first two theorems are presented in Sections 7 and 8.

1. The Rademacher-type results.

Let Γ_{X} denote the space of all integer-valued Radon measures on a Riemannian manifold X with Riemannian inner product g. We will assume throughout this paper that (X, g) is smooth, connected and complete. In the case where X is onedimensional we will assume that X equals ℝ with its Euclidean metric. Throughout this paper, we will assume that Γ_{X} is endowed with the topology of vague convergence. Then Γ_{X} is Polish as a vaguely closed subset of the space of all non-negative Radon measures on X. The set of all γ ∈ Γ_{X} such that γ({x}) ∈ {0, 1} is usually called configuration space over X, but we will also call Γ_{X} itself configuration space.

The space Γ_{X} carries the following geometric structure which was defined in Albeverio, Kondratiev and Röckner (1996), (1997a). The “tangent space” T_{γ}Γ_{X} of Γ_{X} in some γ is given as L^{2}(X → T X, γ), i.e., the space of all sections V in the tangent bundle T X of X which are square-integrable with respect to γ:

∥V ∥_{T_{γ}Γ_{X}}^{2} := ⟨V, V ⟩_{T_{γ}Γ_{X}} := ∫ g_{x}(V, V) γ(dx) < ∞.

For ease of notation, we will also use ∥ · ∥_{γ} and ⟨·, ·⟩_{γ} instead of ∥ · ∥_{T_{γ}Γ_{X}} and ⟨·, ·⟩_{T_{γ}Γ_{X}}. By endowing the tangent space T_{γ}Γ_{X} with the inner product ⟨·, ·⟩_{γ}, Γ_{X} obtains a Riemannian-type structure, which is is non-trivial (i.e., varies with γ) even when the underlying space X is flat.

A suitable space of “smooth test functions” on Γ_{X} is the space FC_{b}^{∞} which consists of all functions u on Γ_{X} of the form

(1.1)  u(γ) = F (⟨f_{1}, γ⟩, …, ⟨f_{n}, γ⟩),  γ ∈ Γ_{X},

for some n ∈ ℕ, F ∈ C_{b}^{∞}(ℝ^{n}), and f_{1}, …, f_{n} ∈ C_{0}^{∞}(X). For u as in (1.1), we define its “gradient” ∇^{Γ}u as a mapping from Γ_{X} × X to T X, i.e., as a section of the tangent bundle T Γ_{X} = ⋃_{γ} T_{γ}Γ_{X}:

(1.2) ∇^{Γ}u(γ, x) := ∑_{i=1}^{n} ∂_{i}F (⟨f_{1}, γ⟩, …, ⟨f_{n}, γ⟩)∇^{X}f_{i}(x),  γ ∈ Γ_{X}, x ∈ X.

Here ∂_{i} means partial derivative in direction of the i-th coordinate, and ∇^{X} is the usual gradient on X. Alternatively, ∇^{Γ}u can be obtained using directional derivatives on Γ_{X}. To this end, let V_{0}(X) denote the space of all smooth vector fields on X having compact support. Then, for fixed γ ∈ Γ_{X}, the flow of diffeomorphisms (ψ_{t})_{t∈ℝ} generated by some V ∈ V_{0}(X) induces a curve ψ_{t}^{∗}γ := γ ∘ (ψ_{t})^{−1}, t ∈ ℝ, on Γ_{X}. With these notations we get that

\frac{d}{dt}|_{t=0}u(ψ_{t}^{∗}γ) = ⟨∇^{Γ}u(γ), V ⟩_{γ} =: ∇_{V}^{Γ} u(γ), γ ∈ Γ_{X}, u ∈ FC_{b}^{∞},

which in particular implies that (1.2) does not depend on a special representation of u as in (1.1).

Let us now introduce an L^{2}-Wasserstein type distance ρ on Γ_{X} as follows.

ρ(ω, γ) := inf \{\sqrt{∫ d(x, y)^{2} η(dx, dy)} | η ∈ Γ_{ω,γ} \}, ω, γ ∈ Γ_{X}, where Γ_{ω,γ} denotes the set of η ∈ Γ_{X×X} having marginals ω and γ, and d is the Riemannian distance function on X. Note that ρ(ω, γ) will be infinite if ω(X) ≠ γ(X), because Γ_{ω,γ} will then be empty. But also if both ω and γ are infinite configurations one will find that ρ(ω, γ) = ∞ in general as the following example shows. Take X = ℝ, ω = ∑_{z∈ℤ} δ_{z}, and γ = ω − δ_{0}, where δ_{z} denotes the Dirac measure in z. Obviously, convergence with respect to ρ implies vague convergence.

Let us now formulate a set of assumptions we will impose in the sequel on a probability measure μ on Γ_{X}.

Assumption 1.1: We suppose that μ is a Borel probability measure on Γ_{X} such that the following conditions hold.

(a) γ({x}) ∈ {0, 1}, for all x ∈ X and μ-a.e. γ.

(b) The mapping γ ↦ γ(K) is in L^{2}(μ), for each compact K ⊂ X.

(c) For any n ∈ ℕ, either μ({γ | γ(X) = n}) > 0 or μ({γ | γ(X) ≥ n}) > 0 corresponding to whether X is compact or non-compact respectively.

(d) For all V ∈ V_{0}(X) and t ∈ ℝ, μ is quasi-invariant with respect to the flow ψ_{t}^{∗} of V, i.e., μ ∘ (ψ_{t}^{∗})^{−1} ≈ μ. Moreover we assume that μ-a.s. ess inf_{r≤s≤t}Φ_{s} > 0, for all finite r < t, where

Φ_{s} := \frac{dμ ∘ (ψ_{V,s}^{∗})^{−1} ⊗ ds}{dμ ⊗ ds}.

(e) μ satisfies the following integration by parts formula. If u, v ∈ FC_{b}^{∞} and V ∈ V_{0}(X), then there is an element ∇_{V}^{Γ∗}v ∈ L^{2}(Γ_{X}, μ) such that

∫ ∇_{V}^{Γ} u v dμ = ∫ u ∇_{V}^{Γ∗}v dμ.

Our main examples of measures satisfying Assumption 1.1 are Ruelle-type Gibbs measures corresponding to a pair potential satisfying certain assumptions (cf. Proposition 2.1 below). They will be discussed in detail in Section 2. Another example is provided by the following class of mixed Poisson measures.

Example 1.2: Let m denote the canonical Riemannian volume element on X and fix a measure σ having a smooth and strictly positive density with respect to m. Then consider a measure μ on Γ_{X} which is given as

μ = ∫_{[0,∞)} π_{s·σ} λ(ds),

where π_{s·σ} denotes the Poisson measure with intensity s · σ (for s = 0, π_{s·σ} will be the Dirac mass on the empty configuration), and λ is a probability measure on [0, ∞) such that ∫ s^{2} λ(ds) < ∞ and λ({0}) < 1. Then we claim that μ satisfies Assumption 1.1. Indeed, (a) and (c) are trivial, (b) follows from our assumptions on λ, (e) was shown in Albeverio, Kondratiev and Röckner (1997a), and quasiinvariance under diffeomorphisms is well-known with an explicit formula for the densities which then implies the positivity in (d) (see e.g. Albeverio, Kondratiev and Röckner (1997a)).

Theorem 1.3: Suppose that μ satisfies Assumption 1.1 and u ∈ L^{2}(μ) is ρ-Lipschitz continuous. Then there exist a measurable subset Γ_{0} of Γ_{X} having full μ-measure and a measurable section ∇^{Γ}u of T Γ_{X} with the following properties.

(i) For all γ ∈ Γ_{0}, ∥∇^{Γ}u(γ)∥_{γ} ≤ Lip(u). (ii) If V ∈ V_{0}(X) is a vector field possessing the flow (ψ_{t})_{t∈ℝ}, then

\frac{u(ψ_{t}^{∗}γ) − u(γ)}{t} ⟶ ⟨∇^{Γ}u(γ), V ⟩_{γ}, as t → 0,

for all γ ∈ Γ_{0} and in L^{2}(μ ∘ (ψ_{s}^{∗})^{−1}), for all s ∈ ℝ.

Theorem 1.3 suggests that ρ-Lipschitz functions in L^{2}(μ) should be contained in a suitable Sobolev space. To make this idea precise, let us introduce the following Dirichlet spaces. Let 𝔽 denote the set of all bounded measurable functions on Γ_{X} for which there exists a measurable section ∇^{Γ}u of T Γ_{X} such that

E_{μ}^{Γ}(u, u) := ∫ ⟨∇^{Γ}u, ∇^{Γ}u⟩ dμ < ∞

and such that

(1.3) \frac{u ∘ ψ_{t}^{∗} − u}{t} ⟶ ⟨∇^{Γ}u, V ⟩  as t → 0 in L^{2}(μ ∘ (ψ_{s}^{∗})^{−1}), for all s ∈ ℝ,

whenever V ∈ V_{0}(X) with flow (ψ_{t})_{t∈ℝ}. By 𝔽^{(c)} we denote the subset of all continuous elements of 𝔽. Clearly FC_{b}^{∞} ⊂ 𝔽^{(c)} by Assumption 1.1 (b).

Proposition 1.4: Assume that μ satisfies Assumption 1.1.

(i) (E_{μ}^{Γ}, FC_{b}^{∞}), (E_{μ}^{Γ}, 𝔽), and (E_{μ}^{Γ}, 𝔽^{(c)}) are closable and their closures, denoted by (E_{μ}^{Γ}, F_{0}), (E_{μ}^{Γ}, F), and (E_{μ}^{Γ}, F^{(c)}) respectively, are Dirichlet forms. Clearly, F_{0} ⊂ F^{(c)} ⊂ F.

(ii) If u ∈ L^{2}(μ) is ρ-Lipschitz continuous, then u ∈ F. (iii) Suppose that μ = ∫ π_{s·σ} λ(ds) is as in Example 1.2 and that the X-valued

Brownian motion having the drift ∇^{X} log(dσ/dm) is conservative. Then F_{0} = F^{(c)} = F.

(iv) For each u ∈ F, there is a section ∇^{Γ}u of the tangent bundle T Γ_{X} such that E_{μ}^{Γ}(u, u) = ∫ ⟨∇^{Γ}u(γ), ∇^{Γ}u(γ)⟩_{γ} μ(dγ) I.e., our Dirichlet form admits a carré du champs operator given by Γ_{E}(u, v)(γ) = ⟨∇^{Γ}u(γ), ∇^{Γ}v(γ)⟩_{γ}.

Proposition 1.4 will be proved in Section 6 below. A priori it is not clear whether one of the identities in (iii) transfer to more general cases than mixed Poisson measures. An investigation in this direction will be the subject of a future work. Our next result first gives a partial converse to Theorem 1.3. Then we show in (ii) that ρ is in fact the largest metric which yields the assertion of this theorem.

Theorem 1.5: Suppose that μ satisfies Assumption 1.1.

(i) If u ∈ F satisfies Γ_{E}(u, u) ≤ C^{2} μ-a.e. and if u has a ρ-continuous μ-version, then there exists a μ-measurable μ-version ũ which is ρ-Lipschitz continuous and satisfies Lip(ũ) ≤ C.

(ii) ρ is the intrinsic metric of the Dirichlet forms (E_{μ}^{Γ}, F) and (E_{μ}^{Γ}, F^{(c)}), i.e.,

ρ(γ, ω) =

(1.4) = sup {u(γ) − u(ω) | u ∈ F ∩ C(Γ_{X}) and Γ_{E}(u, u) ≤ 1 μ-a.e. on Γ_{X}}

= sup {u(γ) − u(ω) | u ∈ 𝔽^{(c)} and Γ_{E}(u, u) ≤ 1 μ-a.e. on Γ_{X}}.

Remark: We believe that in (i) the assumption that u has a ρ-continuous version can be dropped, at least if μ is a mixed Poisson measure. Theorem 1.5 (ii) states in particular that continuous functions u ∈ 𝔽^{(c)} with Γ_{E}(u, u) bounded are already ρ-Lipschitz continuous. However, if X is not compact an arbitrary ρ-Lipschitz continuous u will in general have uncountably many ρ-Lipschitz continuous μ-versions with arbitrarily large Lipschitz constant, which can be seen by modifying u on a single fiber {ρ(γ, ·) < ∞} having measure 0. Therefore, it would not make sense to replace 𝔽^{(c)} or F ∩ C(Γ_{X}) in (1.4) by a larger class of not necessarily continuous functions. On the other hand, it could be useful to know whether 𝔽^{(c)} can be replaced by the smaller set FC_{b}^{∞}. Note that (1.4) is reminiscent of the well-known Kantorovich-Rubinstein theorem for the L^{1}-Wasserstein metric between probability measures. One might guess that a similar variational formula as (1.4) holds for the classical L^{2}-Wasserstein distance on the space of probabilities. It seems that in this case only variational characterizations involving non-symmetric expressions are known to date (cf. Dudley (1989)). If X is compact, then ρ metrizes the vague topology on Γ_{X}. Hence the well-known results on intrinsic distances of regular Dirichlet forms apply (see e.g. Sturm (1995)). However, if X is not compact the situation changes completely, and one is reminded of the Cameron-Martin distance on path space.

2. Applications to Ruelle-type Gibbs measures Let us recall the terminology used for Ruelle measures. Suppose X = ℝ^{d},

and ϕ : ℝ^{d} → ℝ ∪ {+∞} is a pair potential, i.e., ϕ is measurable and satisfies ϕ(x) = ϕ(−x). If Λ ⊂ ℝ^{d} is open bounded and non-empty the conditional energy E_{Λ}^{ϕ} : Γ_{ℝ^{d}} → ℝ ∪ {+∞} is defined as

E_{Λ}^{ϕ}(γ) := ∫_{Λ} ∫_{ℝ^{d}} ϕ(x − y)I{x ≠ y} γ(dx) γ(dy)

if

∫_{Λ} ∫_{ℝ^{d}} |ϕ(x − y)|I{x ≠ y} γ(dx) γ(dy) < ∞,

and E_{Λ}^{ϕ}(γ) := +∞ otherwise. For r = (r^{1}, …, r^{d}) ∈ ℤ^{d}, let Q_{r} denote the cube

Q_{r} := \{x ∈ ℝ^{d} | r^{i} − \frac{1}{2} ≤ x^{i} < r^{i} + \frac{1}{2}\},

and, for N ∈ ℕ define Λ_{N} := [−N, N]^{d}. Then ϕ is called superstable, if there are constants A > 0 and B ≥ 0 such that

E_{Λ_{N}}^{ϕ} (γ_{Λ_{N}}) ≥ ∑_{r∈ℤ^{d}} [Aγ_{Λ_{N}} (Q_{r})^{2} − Bγ_{Λ_{N}} (Q_{r})], for all γ ∈ Γ_{ℝ^{d}} and N ∈ ℕ.

Recall that γ_{Λ} denotes the restriction of γ to Λ. Let | · |_{∞} denote the maximum norm on ℝ^{d}. ϕ is called lower regular if there exists a decreasing positive function a : ℕ → [0, ∞) such that ∑_{r∈ℤ^{d}} a(|r|_{∞}) < ∞ and, for any Λ′, Λ′′ which are finite unions of cubes Q_{r} and disjoint,

\begin{aligned}W (γ_{Λ′}|γ_{Λ′′}) & := ∫_{Λ′} ∫_{Λ′′} ϕ(x − y) γ(dx) γ(dy) \\ & ≥ − ∑_{r′,r′′∈ℤ^{d}} a(|r′ − r′′|_{∞})γ_{Λ′}(Q_{r′})γ_{Λ′′}(Q_{r′′}), \end{aligned}

for all γ ∈ Γ_{ℝ^{d}}. See Ruelle (1970) and the references therein for a discussion of these conditions. Now let m denote Lebesgue measure on ℝ^{d} and let z > 0 be fixed. Then let Z_{Λ}^{ϕ} : Γ_{ℝ^{d}} → [0, ∞] be the partition function

Z_{Λ}^{ϕ}(γ) = ∫ exp ( − E_{Λ}^{ϕ}(γ_{Λ^{C}} + ω_{Λ})) π_{z·m}(dω),

where Λ^{C} := ℝ^{d} \ Λ For B ⊂ Γ_{ℝ^{d}} measurable, we define

Π_{Λ}^{ϕ}(γ, B) = I\{Z_{Λ}^{ϕ} < ∞\}(γ)\frac{1}{Z_{Λ}^{ϕ}(γ)} ∫ IB (γ_{Λ^{C}} +ω_{Λ}) exp (−E_{Λ}^{ϕ}(γ_{Λ^{C}} +ω_{Λ})) π_{z·m}(dω).

The system Π_{Λ}^{ϕ}, Λ ⊂ ℝ^{d} open and bounded, is a specification and μ is a Gibbs measure with respect to Π_{Λ}^{ϕ} if it satisfies the equilibrium equations

μΠ_{Λ}^{ϕ} = μ, for all Λ.

Because of the way our specification was defined, μ is also called grandcanonical Gibbs measure associated with ϕ. Such a measure μ is called tempered if it is supported by S_{∞} := ⋃_{n=1}^{∞} S_{n}, where

S_{n} := {γ ∈ Γ_{ℝ^{d}} | ∀ N ∈ ℕ, ∑_{r∈Λ_{N} ∩ℤ^{d}} γ(Q_{r})^{2} ≤ n^{2}(2N + 1)^{d}}.

According to Section 5 of Ruelle (1970), the set of tempered grandcanonical Gibbs measures is non-empty, provided the potential ϕ is superstable, lower regular, and satisfies the following integrability condition

(2.1)  ∫_{ℝ^{d}} |1 − e^{−ϕ(x)}| dx < +∞.

The following differentiability condition on ϕ was introduced in Albeverio, Kondratiev and Röckner (1997b).

(2.2) e^{−ϕ} is weakly differentiable on ℝ^{d}, ϕ is weakly differentiable on ℝ^{d}\{0}, and the weak gradient ∇ϕ (which is a locally m-integrable function on ℝ^{d}\{0}) considered as an m-a.e. defined function on ℝ^{d} satisfies ∇ϕ ∈ L^{1}(ℝ^{d}, e^{−ϕ}dm) ∩ L^{2}(ℝ^{d}, e^{−ϕ}dm).

Proposition 2.1: Suppose ϕ : ℝ^{d} → ℝ^{d} ∪ {+∞} is a superstable lower regular pair potential with compact support which in addition satisfies (2.1) and (2.2) and which is bounded on any set {x | |x| > r}, for all r > 0. Then every tempered grandcanonical Gibbs measure associated with ϕ (in short: Ruelle measure) satisfies Assumption 1.1.

Together with Proposition 5.6 below this result immediately yields the following corollary.

Corollary 2.2: If ϕ is as in Proposition 2.1, then every Ruelle measure associated with ϕ has full topological support on Γ_{X}.

Proof of Proposition 2.1: Suppose first that ϕ is superstable, lower regular, and satisfies (2.1). Then Corollary 5.3 of Ruelle (1970) states that, for any tempered grandcanonical Gibbs measure μ and any bounded open set Λ ⊂ ℝ^{d}, there exists σ_{Λ} : Γ_{ℝ^{d}} → [0, ∞) such that, for any measurable function F ≥ 0,

∫ F (γ_{Λ}) μ(dγ) = ∫ σ_{Λ}(γ)F (γ_{Λ}) π_{m}(dγ).

Moreover, there are constants c > 0 and d ∈ ℝ such that

σ_{Λ}(γ) ≤ exp (m(Λ) + ∑_{r∈ℤ^{d}} [ − cγ(Λ ∩ Q_{r})^{2} + dγ(Λ ∩ Q_{r})]),

for π_{m}-a.e. γ ∈ Γ_{ℝ^{d}}. In particular, the density σ_{Λ} is bounded above by a constant. It follows immediately from this result that (a) and (b) of Assumption 1.1 are satisfied for such μ. Condition (c) follows from Lemma 2.3 below. Finally, Albeverio, Kondratiev and Röckner (1997b) show that the quasi-invariance and the integration by parts formula of Assumption 1.1 (d) and (e) respectively hold under our assumptions (cf. Lemma 4.2, Theorem 4.3, and Section 5.1 of Albeverio, Kondratiev and Röckner (1997b)). Moreover, if V ∈ V_{0}(ℝ^{d}) has support contained in Λ and generates the flow (ψ_{t})_{t∈ℝ}, then

\frac{dμ ∘ (ψ_{t}^{∗})^{−1}}{dμ} (γ) = exp [E_{Λ}^{ϕ}(γ) − E_{Λ}^{ϕ}(ψ_{−t}^{∗}γ)]\frac{dπ_{z·m} ∘ (ψ_{t}^{∗})^{−1}}{dπ_{z·m}} (γ),

and the positivity condition in (d) holds under our assumptions on ϕ. Thus Proposition 2.1 is proved.

We owe the following lemma to B. Schmuland.

Lemma 2.3: Let μ be grandcanonical Gibbs measure with respect to ϕ. Then

μ({γ | γ(ℝ^{d}) < ∞}) = 0.

Proof: Let 0 denote the empty configuration. First suppose that Z_{Λ}^{ϕ}(0) < ∞, for all bounded open Λ ⊂ ℝ^{d}. Then

μ(\{0\}) = ∫ \frac{1}{Z_{Λ}^{ϕ}(γ)} ∫_{\{ω_{Λ}+γ_{Λ^{C}} =0\}} exp [ − E_{Λ}^{ϕ}(ω_{Λ} + γ_{Λ^{C}})] π_{z·m}(dω) μ(dγ)

= ∫_{\{γ_{Λ^{C}} =0\}} \frac{1}{Z_{Λ}^{ϕ}(γ)} exp[−z · m(Λ)] μ(dγ) (2.3)

= \frac{1}{Z_{Λ}^{ϕ}(0)} exp[−z · m(Λ)] · μ(γ_{Λ^{C}} = 0).

Now

Z_{Λ}^{ϕ}(0) = ∫ exp [ − E_{Λ}^{ϕ}(ω_{Λ})] π_{z·m}(dω) ≥ exp[−z · m(Λ)] · (1 + z · m(Λ)),

because E_{Λ}^{ϕ}(ω_{Λ}) = 0 if ω(Λ) equals 0 or 1. Together with (2.3) this yields

μ(\{0\}) ≤ \frac{1}{1 + z · m(Λ)} ⟶ 0 as Λ ↑ ℝ^{d}.

Plugging this back into (2.3) gives μ(γ_{Λ^{C}} = 0) = 0, for all Λ, which in turn implies that μ(γ(ℝ^{d}) < ∞) = 0.

Next consider the case where Z_{Λ}^{ϕ}(0) = ∞, for large Λ. Then

μ({γ | γ(ℝ^{d}) < ∞}) = ∫ Π_{Λ}^{ϕ}(γ, {ω | γ(Λ^{C}) + ω(Λ) < ∞}) μ(dγ)
= ∫_{{γ(ℝ^{d})<∞}} Π_{Λ}^{ϕ}(γ, {ω | γ(Λ^{C}) + ω(Λ) < ∞}) μ(dγ).

But if γ(ℝ^{d}) < ∞, then Π_{Λ}^{ϕ}(γ, ·) = Π_{Λ}^{ϕ}(0, ·), for Λ large. Also Π_{Λ}^{ϕ}(0, ·) is the zero measure, for Λ so large that Z_{Λ}^{ϕ}(0) = ∞. This proves the lemma.

3. Application: Potential theory on configuration space Let μ satisfy Assumption 1.1, and define, for A ⊂ Γ_{X},

ρ_{A}(γ) := inf {ρ(ω, γ) | ω ∈ A}.

It will be shown in Lemma 4.1 below that ρ_{A} is a measurable function if A is closed.

Proposition 3.1: If K ⊂ Γ_{X} is compact and c ≥ 0, then c ∧ ρ_{K} is an E_{μ}^{Γ}-quasi continuous function in F^{(c)}.

Proof: For ω ∈ Γ_{X} and r > 0, let ρ_{ω,r} denote the function defined in Lemma 4.2 below, and let ρ_{K,r}(γ) = inf{ρ_{ω,r}(γ) | ω ∈ K}. Let ω_{Λ} denote the restriction of a configuration ω to Λ ⊂ X. If F_{r} = {ω̃ | ∃ ω ∈ K such that ω̃_{B_{r}} = ω_{B_{r}}}, then F_{r} is closed, and ρ_{K,r} = ρ_{F_{r}}. Hence ρ_{K,r} is lower semi-continuous by Lemma 4.1 (vii). However, ρ_{K,r} is also upper semi-continuous as infimum over the continuous functions ρ_{ω,r}. Hence ρ_{K,r} is continuous.

Let us now show that lim_{r↑∞} ρ_{K,r}(γ) = ρ_{K}(γ), for all γ ∈ Γ_{X}. We first note that the limit exists since r ↦ ρ_{K,r} is increasing. Furthermore, by Lemma 4.1 there is ω_{γ} ∈ K such that

ρ_{K}(γ) = ρ(ω_{γ}, γ) = lim_{r↑∞} ρ_{ω_{γ},r}(γ) ≥ lim_{r↑∞} ρ_{K,r}(γ),

where we have used Lemma 4.2 (iii) for the second identity. To prove that also ρ_{K}(γ) ≤ lim_{r} ρ_{K,r}(γ) choose α < ρ_{K}(γ). By Lemma 4.2 (iv), the set U_{r} := {ω | ρ_{ω,r}(γ) > α} is open. In addition, Lemma 4.2 (iii) implies that, for any ω ∈ K, there is some r_{ω} ∈ (0, ∞) such that ρ_{ω,r_{ω}}(γ) > α. Thus {U_{r} | r > 0} covers K, and hence there must be some r^{(α)} < ∞ with K ⊂ U_{r^{(α)}}. In other words, ρ_{K,r^{(α)}}(γ) ≥ α. Hence we conclude that ρ_{K}(γ) ≤ lim_{r} ρ_{K,r}(γ).

Next, Theorem 1.3 and Lemma 4.1 (viii) yield that u_{r} := c∧ρ_{K,r} ∈ 𝔽^{(c)} ⊂ F^{(c)} and that E_{μ}^{Γ}(u_{r}, u_{r}) ≤ 1. But this implies the assertion (cf. Ma and Röckner (1992), Section III.3).

Corollary 3.2: If μ(A) = 1, then {ρ_{A} > 0} is exceptional for (E_{μ}^{Γ}, F^{(c)}).

Proof: By inner regularity there are compact sets K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ A such that μ(K_{n}) ↑ 1. Consider the functions u_{n}(γ) := ρ_{K_{n}}(γ) ∧ 1. Then u_{n} ∈ F^{(c)}, and E_{μ}^{Γ}(u_{n}, u_{n}) ≤ 1 by Proposition 3.1, Theorem 1.3, and Lemma 4.1 (viii) below. Therefore u := lim_{n} u_{n} is non-negative and E_{μ}^{Γ}-quasi continuous by Proposition 3.1 and standard arguments (see e.g. Ma and Röckner (1992) Section III.3). In addition u = 0 on ⋃_{n} K_{n}, and hence u = 0 μ-a.s. Hence u = 0 even quasi everywhere by Proposition III.3.9 of Ma and Röckner (1992). But {u = 0} ⊂ {ρ_{A} > 0}.

Let us now look at some applications of Corollary 3.2: Example 3.3:

(i) If A = {γ | γ(X) = ∞} one sees immediately that A = {ρ_{A} < ∞}. Hence μ(A) = 1 implies that A^{C} is exceptional, whereas μ(A) = 0 implies that A is exceptional. this result has first been proved by Byron Schmuland (private communication).

(ii) For the next application, suppose X = ℝ^{d}. Consider the shift transformation θ_{x}, which is defined by θ_{x}γ = δ_{x} ∗ γ. Note that ρ(ω, γ) < ∞ implies that

|u(θ_{x}ω) − u(θ_{x}γ)| ⟶ 0 as |x| → ∞,

for all functions u which are uniformly continuous with respect to a metric δ for the vague topology on Γ_{ℝ^{d}} having the form δ(ω, γ) = ∑_{i=1}^{∞} 2^{−i}| ∫ f_{i} dω − ∫ f_{i} dγ| ∧ 1 with convergence determining f_{i} ∈ C_{0}(ℝ^{d}). Now let

A_{μ} = \{γ ∈ Γ_{ℝ^{d}} | \frac{1}{|V_{n}|} ∫_{V_{n}} δ_{θ_{x}γ} dx ⟶ μ as n ↑ ∞\},

where V_{n} is the box [−n, n]^{d} and convergence is supposed to hold in the weak sense. Then again A_{μ} = {ρ_{A_{μ}} < ∞}. But if μ is ergodic with respect to θ_{x} the spatial ergodic theorem of Nguyen and Zessin (1979) implies μ(A_{μ}) = 1. Thus A_{μ}^{C} is exceptional by our corollary. If μ is not ergodic one can use its ergodic decomposition to get an analogous result.

(iii) A similar reasoning as above applies to the strong law of large numbers.

The above examples exhibit an interesting relation between the tail structure of Γ_{ℝ^{d}} and the σ-field of all events A with the property that

A = {ρ_{A} < ∞}.

This relation and its application to Gibbs measures will be exploited in a future work.

Next we present a short proof for the quasi-regularity of our Dirichlet form (E_{μ}^{Γ}, F^{(c)}). This property implies in particular that the form is associated with a diffusion process having μ as symmetrizing measure. Hence our exceptional sets above can be interpreted as polar sets for this diffusion process. Note that an slightly stronger result, the quasi-regularity of (E_{μ}^{Γ}, F_{0}), has been proved in Ma and Röckner (1997). We refer to Ma and Röckner (1992), Chapter IV, for the terminology below.

Corollary 3.4: (E_{μ}^{Γ}, F^{(c)}) is quasi-regular.

Proof: It suffices to show that Cap is tight. To this end, let K_{1} ⊂ K_{2} ⊂ ⋯ ⊂ Γ_{X} be compact with μ(K_{n}) → 1. Then note that the sets {ρ_{K_{n}} ≤ 1/2} are also compact by Lemma 4.1 (vii) below. But by standard arguments there exists a subsequence (n_{m})_{m∈ℕ} such that u_{N} := N^{−1} ∑_{m=1}^{N} ρ_{K_{nm}} ∧ 1 → 0 wrt E_{μ}^{Γ} + (·, ·)_{L^{2}(μ)}. Moreover, u_{N} ≥ 1/2 on {ρ_{K_{n_{N}}} ≥ 1/2}. Hence

Cap(ρ_{K_{n_{N}}} > 1/2) ≤ Cap(u_{N} > 1/2) ≤ E_{μ}^{Γ}(u_{N}, u_{N}) + ∫ u_{N}^{2} dμ ⟶ 0

by Proposition III.3.4 of Ma and Röckner (1992).

4. Topological properties of ρ In this section, we collect some preliminary results of topological kind concern-

ing our metric ρ. Recall that Γ_{X} is always endowed with the vague topology.

Lemma 4.1: Let π_{i} : X × X → X, i = 1, 2 denote the projection on the i-th coordinate

(i) The mapping Γ_{X×X} ∋ η ↦ ∫ d(x, y)^{2} η(dx, dy) is lower semi-continuous. (ii) If K ⊂ Γ_{X} is compact and i ∈ {1, 2}, then the set {η | π_{i}^{∗}η ∈ K} is relatively

compact in Γ_{X×X}.

(iii) For every α > 0, i = 1, 2, the projection map π_{i}^{∗} : Γ_{X×X} → Γ_{X} restricted to

the closed set G_{α} := {η ∈ Γ_{X×X} | ∫ d(x, y)^{2} η(dx, dy) ≤ α^{2}} is continuous. (iv) Suppose γ, ω ∈ Γ_{X} have finite ρ-distance. Then there is at least one η^{∗} ∈ Γ_{γ,ω}

such that ρ(γ, ω) = ( ∫ d(x, y)^{2} η^{∗}(dx, dy))^{1/2}

(v) The map G_{α} ∋ η ↦ (π_{1}^{∗}η, π_{2}^{∗}η) ∈ Γ_{X} × Γ_{X} is closed (i.e., {(π_{1}^{∗}η, π_{2}^{∗}η) | η ∈ F ∩ G_{α}} is closed in Γ_{X} × Γ_{X}, for all closed F ⊂ Γ_{X×X}). In particular, π_{1}^{∗}, π_{2}^{∗} : G_{α} → Γ_{X} are both closed.

(vi) ρ is lower semi-continuous on Γ_{X} × Γ_{X}. (vii) Let A ⊂ Γ_{X}, A closed. Then ρ_{A} is lower semi-continuous (hence measurable)

on Γ_{X}. If A is compact, then {ρ_{A} ≤ α} is compact, for all α ≥ 0. In particular, closed ρ-balls are compact.

(viii) ρ_{A} ∧ c is ρ-Lipschitz continuous with Lipschitz constant ≤ 1, for all A ⊂ Γ_{X} and c ≥ 0.

Proof: (i) is trivial. For the proof of (ii), let C denote the set under consideration. Then C is relatively compact if and only if sup_{η∈C} η(F) < ∞, for all compact sets F ⊂ X × X. But, for such F,

sup_{η∈C} η(F) ≤ sup_{η∈C} π_{i}^{∗}η(π_{i}(F)) ≤ sup_{γ∈K} γ(π_{i}(F)) < ∞

by compactness of K.

To prove (iii), let f ∈ C_{0}(X), and choose g ∈ C_{0}(X) so that g ≡ 1 on (supp f)^{α} := {d_{suppf} ≤ α}. Then no η ∈ G_{α} charges a point (x, y) where x ∈ supp f and y ∉ (supp f)^{α}. Hence, for all η ∈ G_{α},

∫ f(x) π_{1}^{∗}η(dx) = ∫ f(x) η(dx, dy) = ∫ f(x)g(y) η(dx, dy).

In particular, π_{1}^{∗}η ∈ Γ_{X} and it follows that π_{1}^{∗}, restricted to G_{α}, is continuous.

(iv) follows from (i), (ii), and (iii). For the proof of (v), let F ⊂ Γ_{X×X}, F closed, and η_{k} ∈ F ∩ G_{α}, k ∈ ℕ, such that π_{i}^{∗}η_{k} → γ_{i} ∈ Γ_{X}, i = 1, 2. Then

η_{k} ∈ {η ∈ F ∩ G_{α} | π_{1}^{∗}η ∈ {π_{1}^{∗}η_{k} | k ∈ ℕ} ∪ {γ_{1}}} =: C,

which is a compact set by (i) and (ii). Hence selecting a subsequence if necessary we may assume that η_{k} → η′ ∈ C ⊂ F ∩ G_{α}. Consequently, by continuity γ_{i} = lim_{k} π_{i}^{∗}η_{k} = π_{i}^{∗}η. Hence (γ_{1}, γ_{2}) ∈ {(π_{1}^{∗}η, π_{2}^{∗}η) | η ∈ F ∩ G_{α}}.

(vi). Let α ≥ 0. Then by (iv)

{(γ, ω) ∈ Γ_{X} × Γ_{X} | ρ(ω, γ) ≤ α} = {(π_{1}^{∗}η, π_{2}^{∗}η) | η ∈ G_{α}}.

But by (v) the latter is a closed set.

(vii). Let α ≥ 0 and γ ∈ Γ_{X}. Then

\overline{B}_{α}^{ρ}(γ) := \{ω ∈ Γ_{X} | ρ(γ, ω) ≤ α\} = \{π_{2}^{∗}η | η ∈ G_{α}, π_{1}^{∗}η = γ\}

is compact as the continuous image of {η ∈ G_{α} | π_{1}^{∗}η = γ}. In particular, ω ↦ ρ(γ, ω) is lower semi-continuous, and if r := ρ_{A}(γ) < ∞ there exists ω_{γ} ∈ C := \overline{B}_{r+1}^{ρ}(γ) ∩ A such that ρ(γ, ω_{γ}) = ρ_{C}(γ) = ρ_{A}(γ). Hence

{ρ_{A} ≤ α} = {π_{1}^{∗}η | η ∈ G_{α}, π_{2}^{∗}η ∈ A}.

But the latter set is closed by (v), and even compact if A is by (ii) and (iii). (viii) is trivial, and the lemma is proved.

The next Lemma will in particular imply the quasi-continuity of ρ(ω, ·), for fixed ω. Note that its Assertion (i) needs that X is connected.

Lemma 4.2: Let B_{r} denote the open geodesic ball of radius r > 0 centered in some fixed point of X, and recall that γ_{B_{r}} denotes the restriction of γ ∈ Γ_{X} to B_{r}. For ω ∈ Γ_{X}, we define the closed set A_{ω,r} := {γ ∈ Γ_{X} | γ_{B_{r}} = ω_{B_{r}}}, and let ρ_{ω,r} := ρ_{A_{ω,r}}.

(i) If B_{r} ≠ X, then ρ_{ω,r}(γ) < ∞ if and only if γ(X) ≥ ω(B_{r}). (ii) ρ_{ω,r} is a continuous function from Γ_{X} to [0, ∞]. (iii) ρ_{ω,r}(γ) ↗ ρ(ω, γ) as r ↑ ∞, for all pairs ω, γ ∈ Γ_{X}. (iv) ω ↦ ρ_{ω,r}(γ) is lower semi-continuous if γ is fixed.

Proof: (i). First fix ω ∈ Γ_{X} and write ω_{B_{r}} as ∑_{i=1}^{n} δ_{y^{i}}. If γ ∈ Γ_{X} is given write it as ∑_{i∈I} δ_{z^{i}}, for some index set I ⊂ ℕ. If |I| < n it is clear that ρ_{ω,r}(γ) = ∞. Therefore we can assume |I| ≥ n in the sequel. Then we can find n points x^{1}, …, x^{n} such that γ′ := γ − ∑_{i=1}^{n} δ_{x^{i}} is a non-negative measure. Then we can write γ′_{B_{r}} as ∑_{i=1}^{m} δ_{x^{i+n}}, for some m ≥ 0 and x^{n+1}, …, x^{n+m} ∈ B_{r}. For i = 1, …, m, we then pick an y^{n+i} ∈ ∂B_{r} which realizes the Riemannian distance of x^{n+i} to the boundary ∂B_{r} of B_{r}. If we now define an element ω̃ of Γ_{X} by ω̃ := γ′_{B_{r}^{C}} + ∑_{i=1}^{n+m} δ_{y^{i}} it is clear that ω̃_{B_{r}} = ω_{B_{r}} and that ρ(ω̃, γ)^{2} ≤ r^{2}m + ∑_{i=1}^{n} d(x^{i}, y^{i})^{2} < ∞.

(ii). In view of Lemma 4.1 (vii) it suffices to show upper semicontinuity of γ ↦ ρ_{ω,r}(γ) = ρ_{A_{ω,r}}(γ). To this end, suppose we are given a sequence (γ_{k})_{k∈ℕ} ⊂ Γ_{X} with γ_{k} → γ. If ρ_{ω,r}(γ) = ∞ we are done. Thus assume ρ_{ω,r}(γ) < ∞ in the sequel. Note that, for any ω′ ∈ A_{ω,r} and η ∈ Γ_{X×X} so that

π_{1}^{∗}η = γ, (π_{2}^{∗}η)_{B_{r}} = ω_{B_{r}}, ρ(γ, ω′)^{2} = ∫ d(x, y)^{2} η(dx, dy),

we can construct a new η′ by replacing any (x, y) ∈ supp η with x, y ∈ B_{r}^{C} by (x, x). Then

π_{1}^{∗}η′ = γ, (π_{2}^{∗}η′)_{B_{r}} = ω_{B_{r}}, but ∫ d(x, y)^{2} η′(dx, dy) ≤ ∫ d(x, y)^{2} η(dx, dy).

Since A_{ω,r} is closed, we can hence find some ω^{∗} ∈ A_{ω,r} and η^{∗} ∈ Γ_{γ,ω^{∗}} such that ρ_{ω,r}(γ) = ρ(ω^{∗}, γ), and such that η^{∗} is optimal in the sense of 4.1 (iv) and has the form η^{∗} = ∑_{i=1}^{N} δ_{(x^{i},y^{i})} + ∑ δ_{(x,x)}, for (x^{i}, y^{i}) ∈ B_{r} × Γ_{X} ∪ Γ_{X} × B_{r}. Let γ′ = γ − ∑_{i=1}^{N} δ_{x^{i}}, and write γ′_{∂B_{r}} as ∑_{i=1}^{m} δ_{x^{N+i}}. Then there exist x_{k}^{1}, … x_{k}^{N+m} in γ_{k} such that x_{k}^{i} → x^{i}. Define, for k ∈ ℕ,

η_{k} := ∑_{i=1}^{N} δ_{(x_{k}^{i},y^{i})} + ∑_{i=N+1}^{N+m} δ_{(x_{k}^{i},x^{i})} + ∑ δ_{(x_{k},x_{k})},

where third sum is determined by π_{1}^{∗}η_{k} = γ_{k}. Then, for large k, (π_{2}^{∗}η_{k})_{B_{r}} = ω_{B_{r}}, since (γ_{k} − ∑_{i=1}^{N+m} δ_{x_{k}^{i}})(\overline{B}_{r}) = 0 eventually. Therefore,

lim sup_{k↑∞} ρ_{ω,r}(γ_{k})^{2} ≤ lim sup_{k↑∞} ( ∑_{i=1}^{N} d(x_{k}^{i}, y^{i})^{2} + ∑_{i=N+1}^{N+m} d(x_{k}^{i}, x^{i})^{2})

= ∑_{i=1}^{N} d(x^{i}, y^{i})^{2} = ρ_{ω,r}(γ)^{2}.

(iii). It suffices to show that α := sup_{r} ρ_{ω,r}(γ) ≥ ρ(ω, γ). If α = ∞ we are done. Otherwise we know from the proof of (ii) that there are ω_{r}^{∗} such that (ω_{r}^{∗})_{B_{r}} = ω_{B_{r}} and ρ_{ω,r}(γ) = ρ(ω_{r}^{∗}, γ). But ω_{r}^{∗} → ω as r ↑ ∞, since ∫ f dω_{r}^{∗} = ∫ f dω, for all continuous f with support in B_{r}. Thus ρ(ω, γ) ≤ α follows from the lower semicontinuity of ρ(·, γ).

(iv). Let (ω_{n})_{n∈ℕ} be a given sequence converging to ω in Γ_{X}. We have to show that ρ_{ω,r}(γ) ≤ α if there exists an α < ∞ such that sup_{n} ρ_{ω_{n},r}(γ) ≤ α. As above it follows that there are ω_{n}^{∗} with (ω_{n}^{∗})_{B_{r}} = (ω_{n})_{B_{r}} and ρ_{ω_{n},r}(γ) = ρ(ω_{n}^{∗}, γ) ≤ α. By Lemma 4.1 {ω_{n}^{∗} | n ∈ ℕ} is relatively compact and any accumulation point ω^{∗} satisfies ρ(ω^{∗}, γ) ≤ α. Moreover, if f is a continuous function with compact support in B_{r},

∫ f dω^{∗} = lim_{n↑∞} ∫ f dω_{n}^{∗} = lim_{n↑∞} ∫ f dω_{n} = ∫ f dω.

Thus ω_{B_{r}}^{∗} = ω_{B_{r}}, and (iv) is proved.

5. The ρ-geometry on Γ_{X} In this section, we will derive several auxiliary lemmas of geometric kind, which

are needed in order to prove our theorems. One of the key results in this section will be Proposition 5.4 below, which roughly states that the images of some γ ∈ Γ_{X} under V_{0}(X)-flows are a ρ-dense subset of the set of all ω ∈ Γ_{X} which have finite ρdistance with respect to γ. This will in particular imply that any measure satisfying Assumption 1.1 has full support (cf. Proposition 5.6), a property which might be of independent interest in case of the Gibbs measures of Section 2.

Suppose we are given a ρ-continuous path ξ : ℝ → Γ_{X} and two real numbers a and b with a < b. Then, as usual, we define the ρ-energy of ξ in the interval [a, b] as follows.

E_{a,b}(ξ) = sup \{\frac{1}{2} ∑_{i=1}^{n} \frac{ρ(ξ_{t_{i}}, ξ_{t_{i−1}})^{2}}{t_{i} − t_{i−1}} | a = t_{0} < t_{1} < ⋯ < t_{n} = b, n ∈ ℕ\}.

Lemma 5.1: If V ∈ V_{0}(X) has the flow (ψ_{t})_{t∈ℝ} and ξ_{t} is given by ξ_{t} = ψ_{t}^{∗}γ, for some fixed γ ∈ Γ_{X}, then the path ξ is ρ-continuous, and

E_{a,b}(ξ) = \frac{1}{2} ∫_{a}^{b} ∥V ∥_{ξ_{t}}^{2} dt,

for all a < b.

Proof: To prove ‘≤’ we remark that

ρ(ξ_{s}, ξ_{t})^{2} ≤ ∫ d(ψ_{s}(x), ψ_{t}(x))^{2}γ(dx) ≤ ∫ max_{s≤r≤t} g_{ψ_{r}(x)}(V, V)(t − s)^{2} γ(dx).

Now choose an ordered partition Δ = {t_{0}, …, t_{n}} of [a, b]. Then

∑_{i=1}^{n} \frac{ρ(ξ_{t_{i}}, ξ_{t_{i−1}})^{2}}{t_{i} − t_{i−1}} ≤ ∫ ∑_{i=1}^{n} max_{t_{i−1}≤r≤t_{i}} g_{ψ_{r}(x)}(V, V) (t_{i} − t_{i−1}) γ(dx).

The latter converges to

∫ ∫_{a}^{b} g_{ψ_{t}(x)}(V, V) dt γ(dx) = ∫_{a}^{b} ∫ g(V, V) dξ_{t} dt

as the mesh of Δ tends to 0. Let us now prove that E_{a,b}(ξ) ≥ \frac{1}{2} ∫_{a}^{b} ∥V ∥_{ξ_{t}}^{2} dt. To this end, let E_{a,b}^{X}(c_{x}) denote the usual Riemannian energy of the X-valued curve c_{x}(t) := ψ_{t}(x), t ∈ ℝ, over the interval [a, b]. Then, if ε > 0 is given, there is some n = n(x) ∈ ℕ such that, for t_{i} := a + (b − a) · i2^{−n},

\frac{1}{2} ∫_{a}^{b} g_{c_{x}(t)}(V, V) dt = E_{a,b}^{X}(c_{x}) ≤ \frac{1}{2} ∑_{i=1}^{2^{n}} \frac{d(c_{x}(t_{i}), c_{x}(t_{i−1}))^{2}}{t_{i} − t_{i−1}} + ε.

Since V has compact support, n can even be chosen uniformly in x ∈ supp V ∩supp γ. Moreover, it is easy to see that, for large n,

ρ(ξ_{t_{i}}, ξ_{t_{i−1}}) = ( ∫ d(c_{x}(t_{i}), c_{x}(t_{i−1}))^{2} γ(dx))^{1/2}.

Thus

E_{a,b}(ξ) ≥ \frac{1}{2} ∑_{i=1}^{2^{n}} \frac{ρ(ξ_{t_{i}}, ξ_{t_{i−1}})^{2}}{t_{i} − t_{i−1}} ≥ \frac{1}{2} ∫_{a}^{b} ∥V ∥_{ξ_{t}} dt − ε · γ(supp V)

This proves the assertion.

Lemma 5.2: Suppose ξ is as in Lemma 5.1 and u : Γ_{X} → ℝ is ρ-Lipschitz continuous. Then t ↦ u(ξ_{t}) is absolutely continuous and

|\frac{d}{dt}u(ξ_{t})| ≤ Lip(u) · ∥V ∥_{ξ_{t}} for almost every t.

Proof: If Δ = {t_{0}, t_{1}, …, t_{n}} is an ordered partition of some finite interval [a, b], then

∑_{i=1}^{n} \frac{(u(ξ_{t_{i}}) − u(ξ_{t_{i−1}}))^{2}}{t_{i} − t_{i−1}} ≤ 2 Lip(u)^{2}E_{a,b}(ξ) = Lip(u)^{2} ∫_{a}^{b} ∥V ∥_{ξ_{t}}^{2} dt.

The lemma in Chapter II, No. 36 of Riesz and Nagy (1956) states that, under this condition, t ↦ u(ξ_{t}) is absolutely continuous, and that

∫_{a}^{b} (\frac{d}{dt}u(ξ_{t}))^{2} dt ≤ Lip(u)^{2} ∫_{a}^{b} ∥V ∥_{ξ_{t}}^{2} dt.

Thus the lemma is proved, because a and b were arbitrary.

Note that smoothness of (X, g) is essential in the proof of the following lemma, and in fact the assertion may fail if X is only a Lipschitz manifold.

Lemma 5.3: Let γ, ω, and η^{∗} be as in Lemma 4.1 (iv). Suppose that (x_{1}, y_{1}) and (x_{2}, y_{2}) are two points in the support of η^{∗} such that x_{1}, y_{1}, x_{2}, and y_{2} are mutually distinct. Let τ_{i} denote d(x_{i}, y_{i}), and assume that c_{i} : [0, τ_{i}] → X is a minimal geodesic connecting x_{i} with y_{i}, and suppose that c_{i} is parameterized by arc length (i = 1, 2). Then neither c_{1}([0, τ_{1}]) ⊂ c_{2}([0, τ_{2}]) nor c_{2}([0, τ_{2}]) ⊂ c_{1}([0, τ_{1}]) can occur and one of the two following cases holds.

(i) There exists a local geodesic c extending both c_{1} and c_{2}. (ii) c_{1}([0, τ_{1}]) ∩ c_{2}([0, τ_{2}]) = ∅.

Proof: Suppose that c_{1}([0, τ_{1}]) ⊂ c_{2}([0, τ_{2}]). But then we must have that

(5.1)  d(x_{1}, y_{2})^{2} + d(x_{2}, y_{1})^{2} < d(x_{1}, y_{1})^{2} + d(x_{2}, y_{2})^{2},

which contradicts the minimality of η^{∗}.

Now suppose that x_{1}, y_{1}, x_{2}, and y_{2} do not lie on a single local geodesic, but c_{1}([0, τ_{1}]) ∩ c_{2}([0, τ_{2}]) ≠ ∅. Then there are t_{1} ∈ (0, τ_{1}) and t_{2} ∈ (0, τ_{2}) such that c_{1}(t_{1}) = c_{2}(t_{2}). Define two piecewise smooth curves c_{12} and c_{21} by

c_{ij}(t) = \{ \begin{matrix}c_{i}(t) & if t ∈ [0, t_{i}], \\ c_{j}(t − t_{i} + t_{j}) & if t ∈ [t_{i}, t_{i} + τ_{j} − t_{j}]. \end{matrix}

Then the total energy of c_{12} and c_{21} is the same as that of c_{1} and c_{2}, i.e.,

(5.2)  E^{X}(c_{12}) + E^{X}(c_{21}) = E^{X}(c_{1}) + E^{X}(c_{2}) = d(x_{1}, y_{1})^{2} + d(x_{2}, y_{2})^{2},

if E^{X} denotes the energy functional acting on piecewise smooth X-valued curves. Now observe that the tangent vectors of c_{1} and c_{2} cannot be proportional, because c_{1} and c_{2} cannot be extended to one single geodesic. Therefore our curves c_{12} and c_{21} are continuous but not differentiable in t_{1} and t_{2} respectively. But this implies that they cannot be energy-minimizing in the class of curves which are parameterized by arc length and have given endpoints. Hence there are two curves c_{12}^{∗} and c_{21}^{∗} parameterized by arc length connecting x_{1} and y_{2}, and x_{2} and y_{1} respectively such that E^{X}(c_{ij})^{∗} < E^{X}(c_{ij}). But in view of (5.2) this implies that (5.1) also holds in this case. So, again, we arrive at a contradiction to the minimality of η^{∗}.

Proposition 5.4: Suppose ε > 0 is given and ω, γ ∈ Γ_{X} are such that ρ(ω, γ) < ∞. Assume furthermore that γ has the property that

(5.3)  γ({x}) ∈ {0, 1},  for all x ∈ X.

Then, if dim X ≥ 2, there is a vector field V ∈ V_{0}(X) with flow (ψ_{t})_{t∈ℝ} such that ρ(ψ_{1}^{∗}γ, ω) < ε and ∥V ∥_{ψ_{t}^{∗}γ} = ρ(ψ_{1}^{∗}γ, γ), for all t ∈ [0, 1].

If X = ℝ, then there are n ∈ ℕ, t_{i} ≥ 0, and V_{i} ∈ V_{0}(X) with corresponding flows (ψ_{i,t})_{t∈ℝ}, i = 1, …, n, such that t_{1} + ⋯ + t_{n} = 1 and such that

ϕ_{t} := ψ_{i,s} ∘ ψ_{i−1,t_{i−1}} ∘ ⋯ ∘ ψ_{1,t_{1}}, for t = s + t_{i−1} + ⋯ + t_{1}, 0 ≤ s ≤ t_{i},

satisfies ρ(ϕ_{1}^{∗}γ, ω) < ε and ρ(ϕ_{1}^{∗}γ, γ) ≤ ∥V_{i}∥_{ϕ_{t}^{∗}γ} ≤ ρ(ϕ_{1}^{∗}γ, γ) + ε, for t as above.

Proof: First note that we can assume without loss of generality that also ω satisfies (5.3), since we can otherwise alter the corresponding points in the support of ω by an arbitrarily small portion of ε. Choose r > 0 so that the function ρ_{ω,r} of Lemma 4.2 satisfies ρ_{ω,r}(γ) ≥ ρ(ω, γ) − ε. As in the proof of 4.2 (iii) there exist ω^{∗} and η^{∗} ∈ Γ_{γ,ω^{∗}} such that ω_{B_{r}}^{∗} = ω_{B_{r}}, ρ_{ω,r}(γ) = ρ(ω^{∗}, γ), and η^{∗} is optimal in the sense of 4.1 (iv) and has the form ∑_{i=1}^{N} δ_{(x_{i},y_{i})} + ∑ δ_{(x,x)} with x_{i} ≠ y_{i}. Now we choose minimal geodesics c_{i} : [0, 1] → X parameterized by arc length such that c_{i}(0) = x_{i} and c_{i}(1) = y_{i}, i = 1, …, N.

First consider the case dim X ≥ 2. Then we can assume without loss of generality that

(5.4)  c_{i}([0, 1]) ∩ c_{j}([0, 1]) = ∅  for i ≠ j,

because otherwise the situation of (i) of Lemma 5.3 must occur, and we could alter the corresponding points by a small amount to arrive at a configuration which satisfies (5.4). Now we can define V to be ċ_{i}(t) in the points c_{i}(t), for 0 ≤ t ≤ 1 and i = 1, …, N. Then the vector field V is well defined due to (5.4), and we can extend it to an element in V_{0}(X) by standard arguments. Let (ψ_{t})_{t∈ℝ} be its flow. Then by construction

ψ_{t}(x_{i}) = c_{i}(t),  ψ_{1}^{∗}γ = ω^{∗},  and  ∫ g(V, V) dψ_{t}^{∗}γ = ρ(γ, ω^{∗})^{2},

for all t ∈ [0, 1] and i = 1, …, N.

Next consider the case where X = ℝ. Then we are always in the situation of Lemma 5.3 (i), and we can no longer assume (5.4). However, we know that it cannot occur that c_{i}([0, 1]) ⊂ c_{j}([0, 1]), for some i ≠ j, and no point x such that (x, x) ∈ supp η^{∗} can be contained in ⋃_{i} c_{i}([0, 1]). Moreover, if c_{i}([0, 1]) ∩ c_{j}([0, 1]) ≠ ∅, then c_{i} and c_{j} have the same orientation, i.e., (x_{i} − y_{i})(x_{j} − y_{j}) > 0. Now rearrange the x_{i}s such that (x_{1}, y_{1}), …, (x_{m}, y_{m}) are precisely those pairs with x_{i} < y_{i} and such that x_{1} < x_{2} < ⋯ < x_{m} and x_{m+1} > x_{m+2} > ⋯ > x_{N}. Then define a piecewise constant vector field Ṽ_{1} as y_{i} − x_{i} on [x_{i}, y_{i} ∧ x_{i+1}), i = 1, …, m, and y_{i} − x_{i} on (y_{i} ∨ x_{i+1}, x_{i}], and Ṽ_{1} ≡ 0 elsewhere. Let (ψ̃_{1,t})_{t∈ℝ} denote its flow, and define

t̃_{1} := inf {t > 0 | ∃ i, j such that c_{i}(t) = x_{j}} ∧ 1.

By construction ψ̃_{1,t}^{∗}γ is an energy minimizing ρ-geodesic on [0, t̃_{1}] and

ρ(γ, ω^{∗}) = ρ(γ, ψ̃_{1,t̃_{1}}^{∗}γ) + ρ(ψ̃_{1,t̃_{1}}^{∗}γ, ω^{∗}).

In particular,

η_{1}^{∗} := ∫ δ_{(ψ̃_{1,t̃_{1}} (x),y)} η^{∗}(dx, dy)

is again optimal in the sense of 4.1 (iv). Hence we can replace γ by ψ̃_{1,t̃_{1}}^{∗}γ and η^{∗} by η_{1}^{∗}, and start the above construction over again. Then we get a piecewise constant vector field Ṽ_{2} with flow (ψ̃_{2,t})_{t∈ℝ} and some t̃_{2} ∈ (0, 1 − t̃_{1}] as above. It is easy to see that the above algorithm stops at some finite n, i.e., t̃_{1} + ⋯ + t̃_{n} = 1 and ω^{∗} = ψ̃_{n,t̃_{n}}^{∗} ∘ ⋯ ∘ ψ̃_{1,t̃_{1}}^{∗}γ. Applying a mollifier to Ṽ_{1}, …, Ṽ_{n} gives the result.

Let ∥ · ∥_{∞} denote the sup-norm of a function on X, and extend this norm to vector fields V = (V^{1}, …, V^{d}) by setting

(5.5)  ∥V ∥_{∞} := sup_{x∈X} \sqrt{g_{x}(V, V)}.

Lemma 5.5: If V, W ∈ V_{0}(X), (ψ_{t})_{t∈ℝ} and (ϕ_{t})_{t∈ℝ} denote the corresponding flows, and A = supp V ∪ supp W. Then there is a constant c = c(V, A) such that

ρ(ψ_{t}^{∗}γ, ϕ_{t}^{∗}γ) ≤ ct · e^{ct} · ∥V − W ∥_{∞} · \sqrt{γ(A)}, for all γ ∈ Γ_{X} and t ≥ 0.

Proof: By the Nash (1956) embedding theorem X can be isometrically embedded into some ℝ^{n}. Below | · |_{1} will denote the ℓ^{1}-norm on ℝ^{n}. Then

\begin{aligned}|ψ_{t}(x) − ϕ_{t}(x)|_{1} & ≤ ∫_{0}^{t} |V (ψ_{s}(x)) − W (ϕ_{s}(x))|_{1} ds \\ & ≤ (t · sup_{y∈X} |V (y) − W (y)|_{1} + L_{V} · ∫_{0}^{t} |ψ_{s}(x) − ϕ_{s}(x)|_{1} ds) · IA(x), \end{aligned}

where, for V = (V^{1}, …, V^{n}),

L_{V} := ∑_{i=1}^{n} sup \{\frac{|V^{i}(y) − V^{i}(z)|}{|z − y|_{1}} | z ≠ y, z, y ∈ X\}.

Gronwall’s lemma now yields

|ψ_{t}(x) − ϕ_{t}(x)|_{1} ≤ t · sup_{y∈X} |V (y) − W (y)|_{1} · e^{tL_{V}} · IA(x).

Since A is compact, there are positive constants c_{1} and c_{2} depending only on A such that, for all y, z ∈ A and every tangent vector U ∈ T_{y}X,

\frac{1}{c_{1}} |U|_{1} ≤ \sqrt{g_{y}(U, U)} ≤ c_{1}|U|_{1}  and  d(y, z) ≤ c_{2}|y − z|_{1}.

By taking η = ∫ δ_{(ψ_{t}(x),ϕ_{t}(x))} γ(dx) the assertion follows.

Proposition 5.6: Suppose that μ is a probability measure on Γ_{X} which satisfies Assumptions 1.1 (a), (c), and the quasi-invariance of (d). Then μ has full support.

Proof: We only give the proof in the case where X is non-compact and dim X ≥ 2. Only minor modifications are necessary to handle the other cases. Suppose that μ does not have full support. Then we can find an open geodesic ball B_{r} ⊂ X, δ > 0, n ∈ ℕ, and y_{1}, …, y_{n} ∈ B_{r} such that the open set

U := {ω ∈ Γ_{X} | ω(∂B_{r}) = 0 and ω_{B_{r}} = ∑_{i=1}^{n} δ_{x_{i}} with ∑_{i=1}^{n} d(x_{i}, y_{i})^{2} < δ}

satisfies μ(U) = 0, because these sets form a neighborhood base for Γ_{X}. Now choose a compact set

K ⊂ {γ ∈ Γ_{X} | γ(X) ≥ n and γ({x}) ∈ {0, 1} for all x}

with μ(K) > 0. Such a set exists by Assumption 1.1 (a) and (c). Fix γ ∈ K. By Lemma 4.2 we must have ρ_{ω,r}(γ) < ∞, for all ω ∈ U. In particular, there is ω ∈ U with ρ(γ, ω) < ∞. Since U is open, Proposition 5.4 implies that there is even a vector field V_{γ} ∈ V_{0}(X) such that its flow (ψ_{γ,t})_{t∈ℝ} satisfies ψ_{γ,1}^{∗}γ ∈ U. Hence K is covered by (ψ_{γ,1}^{∗})^{−1}U, γ ∈ K, and we can extract a finite subcover (ψ_{1,1}^{∗})^{−1}U, …, (ψ_{n,1}^{∗})^{−1}U. But by quasi-invariance μ((ψ_{i,1}^{∗})^{−1}U) = 0 which contradicts μ(K) > 0.

6. Dirichlet forms on Γ_{X}.

In this section, we prove Proposition 1.4. The following two lemmas will also be needed for the proofs our main results. We suppose in this section that μ satisfies Assumption 1.1.

Lemma 6.1: Let V ∈ V_{0}(X) and denote its flow by (ψ_{t})_{t∈ℝ}. Then

(6.1)  ∫ (u ∘ ψ_{t}^{∗} − u)v dμ = ∫_{0}^{t} ∫ (u ∘ ψ_{s}^{∗}) ∇_{V}^{Γ∗}v dμ ds,

for all bounded and measurable u and all v ∈ FC_{b}^{∞}.

Proof: By a monotone class argument it suffices to prove (6.1) for u ∈ FC_{b}^{∞}. Then also u ∘ ψ_{s}^{∗} ∈ FC_{b}^{∞}. Note that

u ∘ ψ_{t}^{∗} − u = ∫_{0}^{t}(∇_{V}^{Γ} u) ∘ ψ_{s}^{∗} ds = ∫_{0}^{t} ∇_{V}^{Γ} (u ∘ ψ_{s}^{∗}) ds.

Therefore, by Assumption 1.1 (e),

∫ (u ∘ ψ_{t}^{∗} − u)v dμ = ∫_{0}^{t} ∫ (u ∘ ψ_{s}^{∗}) ∇_{V}^{Γ∗}v dμ ds.

This proves the lemma.

Lemma 6.2: If u ∈ 𝔽 and (ψ_{t})_{t∈ℝ} is the flow of some V ∈ V_{0}(X), then

u ∘ ψ_{t}^{∗} − u = ∫_{0}^{t}⟨∇^{Γ}u, V ⟩ ∘ ψ_{s}^{∗} ds, for all t ∈ ℝ, μ-a.e.,

and, for all v ∈ FC_{b}^{∞} and all s ∈ ℝ,

(6.2)  ∫ ⟨∇^{Γ}u, V ⟩ ∘ ψ_{s}^{∗} · v dμ = ∫ (u ∘ ψ_{s}^{∗})∇_{V}^{Γ∗}v dμ.

Proof: Let v ∈ FC_{b}^{∞}. Then by definition of 𝔽 and Lemma 6.1

∫ ⟨∇^{Γ}u, V ⟩ ∘ ψ_{s}^{∗} · v dμ = lim_{t→0} ∫ \frac{1}{t}(u ∘ ψ_{s+t}^{∗} − u ∘ ψ_{s}^{∗})v dμ

= lim_{t→0} \frac{1}{t} ∫_{0}^{t} ∫ (u ∘ ψ_{s+r}^{∗})∇_{V}^{Γ∗}v dμ dr

= ∫ (u ∘ ψ_{s}^{∗})∇_{V}^{Γ∗}v dμ,

because t ↦ u ∘ ψ_{t}^{∗} ∈ L^{2}(μ) is continuous. Hence, again by Lemma 6.1

∫ (u ∘ ψ_{t}^{∗} − u)v dμ = ∫ [ ∫_{0}^{t}⟨∇^{Γ}u, V ⟩ ∘ ψ_{s}^{∗} ds]v dμ.

By continuity of t ↦ u ∘ ψ_{t}^{∗} ∈ L^{2}(μ) the assertion follows.

Following Albeverio, Kondratiev and Röckner (1997b) and Eberle (1995), we let VFC_{b}^{∞} denote the set of all “smooth vector fields” on Γ_{X}, i.e. the set of all sections Y of T Γ_{X} which are of the form Y (γ, x) = ∑_{i=1}^{n} v_{i}(γ) V_{i}(x), for n ∈ ℕ, v_{i} ∈ FC_{b}^{∞}, and V_{i} ∈ V_{0}(X). Then Assumption 1.1 (e) implies that, for u ∈ FC_{b}^{∞},

∫ ⟨∇^{Γ}u(γ), Y (γ)⟩_{γ} μ(dγ) = − ∫ u div _{μ}^{Γ}Y dμ,

where

div _{μ}^{Γ}Y (γ) := − ∑_{i=1}^{n} ∇_{V_{i}}^{Γ∗}v_{i}(γ).

Then (div _{μ}^{Γ}, VFC_{b}^{∞}) is a densely defined linear operator from L^{2}(Γ_{X} → T Γ_{X}, μ) to L^{2}(μ), and we denote its adjoint by (d^{μ}, W^{1,2}). Functions u ∈ W^{1,2} are weakly differentiable in the sense that ∫ ⟨d^{μ}u, Y ⟩ dμ = − ∫ u div _{μ}^{Γ}Y dμ holds for all Y ∈ VFC_{b}^{∞}.

Lemma 6.3: 𝔽 ⊂ W^{1,2} and ∇^{Γ} = d^{μ} on 𝔽.

Proof: Let u ∈ 𝔽 and Y = ∑_{i=1}^{n} v_{i}V_{i} ∈ VFC_{b}^{∞}. Then we get by using (6.2) for s = 0

∫ ⟨∇^{Γ}u, Y ⟩ dμ = ∑_{i=1}^{n} ∫ u∇_{V_{i}}^{Γ∗}v_{i} dμ = − ∫ u div _{μ}^{Γ}Y dμ.

Hence the lemma follows.

Lemma 6.4: Suppose (a, b) ∋ t ↦ v(t) ∈ L^{2}(μ) is differentiable at t_{0} ∈ (a, b) and let φ ∈ C_{b}^{1}(ℝ). Then (a, b) ∋ t ↦ φ(v(t)) ∈ L^{2}(μ) is differentiable at t_{0} with derivative

\frac{d}{dt}|_{t=t_{0}}φ(v(t)) = φ′(v(t_{0}))\frac{dv(t)}{dt}|_{t=t_{0}}.

Proof: Let Δ(t) := (φ(v(t)) − φ(v(t_{0})))/(t − t_{0}). It suffices to show that every sequence t_{n} → t_{0} has a subsequence (t_{n_{k}}) such that Δ(t_{n_{k}}) → φ′(v(t_{0}))(dv(t)/dt)|_{t=t_{0}} in L^{2}(μ). But this follows easily by noting that {(Δ(t_{n}))^{2} | n ∈ ℕ} is uniformly integrabel due to the Lipschitz property of φ and by taking (t_{n_{k}}) so that μ-a.e. (v(t_{n_{k}}) − v(t_{0}))/(t_{n_{k}} − t_{0}) → (dv(t)/dt)|_{t=t_{0}}.

Proof of Proposition 1.4: (i). Since (div _{μ}^{Γ}, VFC_{b}^{∞}) is densely defined, the form Ê(u, u) := ∫ ⟨d^{μ}u, d^{μ}u⟩ dμ, u ∈ W^{1,2}, is closed. Therefore (E_{μ}^{Γ}, FC_{b}^{∞}), (E_{μ}^{Γ}, 𝔽), and (E_{μ}^{Γ}, 𝔽^{(c)}) are all closable by Lemma 6.3 and Proposition I.3.5 of Ma and Röckner (1992). Thus (E_{μ}^{Γ}, F_{0}), (E_{μ}^{Γ}, F), and (E_{μ}^{Γ}, F^{(c)}) are symmetric closed forms, and it remains to check the contraction property to conclude that they are Dirichlet forms. But, for u ∈ 𝔽, this is clear from (1.3) and Lemma 6.4, and if u ∈ F is arbitrary it then follows from Proposition II.4.10 of Ma and Röckner (1992).

Assertion (ii) is immediately implied by Theorem 1.3 if u is bounded. The extension to u ∈ L^{2}(μ) then follows easily by approximating u by (−n) ∨ u ∧ n.

As for (iii), observe that the linear operators ∇^{Γ} and d^{μ} and hence also div _{μ}^{Γ}∇^{Γ} and div _{μ}^{Γ}d^{μ} coincide on FC_{b}^{∞}. But if μ is a mixed Poisson measure as in the assertion, then due to Theorem 4.2 of Albeverio, Kondratiev and Röckner (1997a) and its proof we must have that F = F^{(c)} = F_{0}.

(iv). For u ∈ F, there is a sequence (u_{n})_{n∈ℕ} ⊂ 𝔽 converging to u in F. Thus, if Λ denotes the σ-finite measure Λ(dx, dγ) = γ(dx) μ(dγ), (∇^{Γ}u_{n})_{n∈ℕ} is a Cauchy sequence in L^{2}(X × Γ_{X} → T X, Λ). Hence there exists an element ∇^{Γ}u ∈ L^{2}(X × Γ_{X} → T X, Λ) such that E_{μ}^{Γ}(u, u) = ∫ g_{x}(∇^{Γ}u(γ), ∇^{Γ}u(γ)) Λ(dx, dγ).

7. Proof of Theorem 1.3 Let u ∈ L^{2}(μ) be ρ-Lipschitz continuous with L := Lip(u). Fix V ∈ V_{0}(X)

having the flow (ψ_{t})_{t∈ℝ}. Then consider the set

Ω_{V} := \{γ ∈ Γ_{X} | \frac{u(ψ_{t}^{∗}γ) − u(γ)}{t} → G_{V} (γ) as t → 0, and |G_{V} (γ)| ≤ L · ∥V ∥_{γ}\}.

Then Ω_{V} is measurable because the existence of lim_{t→0} \frac{1}{t}(u(ψ_{t}^{∗}γ) − u(γ)) is equivalent

to the existence of the limit of \frac{1}{r}(u(ψ_{r}^{∗}γ) − u(γ)) for r → 0, r ∈ ℚ, because t ↦ u(ψ_{t}^{∗}γ) is continuous by Lemma 5.2. Moreover, we claim that Ω_{V} has full μ-measure. Indeed, we know by Lemma 5.2 that, for all γ ∈ Γ_{X}, the set of s ∈ ℝ for which ψ_{s}^{∗}γ ∈ Ω_{V} has full Lebesgue measure. Thus

0 = ∫_{0}^{1} ds ∫ μ(dγ) IΩ_{V}^{C} (ψ_{s}^{∗}γ) = ∫_{0}^{1} ds ∫_{Ω_{V}^{C}} μ(dγ) \frac{dμ ∘ (ψ_{s}^{∗})^{−1}}{dμ} (γ).

But, for every s, dμ ∘ (ψ_{s}^{∗})^{−1}/dμ is μ-a.s. strictly positive by Assumption 1.1 (d), and hence μ(Ω_{V}^{C}) = 0.

We now claim that

(7.1)  \frac{u ∘ ψ_{t}^{∗} − u}{t} ⟶ G_{V}  in L^{2}(μ ∘ (ψ_{s}^{∗})^{−1}) as t → 0, for all s ∈ ℝ.

Indeed, we already know from Lemma 5.2 that

(7.2)

sup_{−1≤t≤1} |\frac{u(ψ_{t}^{∗}γ) − u(γ)}{t}| ≤ sup_{−1≤t≤1} \frac{L}{t} ∫_{0}^{t} ∥V ∥_{ψ_{s}^{∗}γ} ds

≤ L(∫ sup_{−1≤s≤1} [g(V, V) ∘ ψ_{s}] dγ)^{1/2}

≤ L∥V ∥_{∞}\sqrt{γ(supp V)}.

By Assumption 1.1 (b), (7.1) thus follows from dominated convergence.

Observe that it follows from (7.2) that (6.1) holds for u even though u is not necessarily bounded. In view of (7.1), this implies that

∫ G_{V} v dμ = ∫ u ∇_{V}^{Γ∗}v dμ, for all v ∈ FC_{b}^{∞},

because s ↦ u(ψ_{s}^{∗}γ) is continuous by Lemma 5.2. Then note that V ↦ ∇_{V}^{Γ∗}v is linear by Assumption 1.1 (e). Hence if V can be written as α_{1}V_{1} + ⋯ + α_{k}V_{k} with α_{i} ∈ ℝ and V_{i} ∈ V_{0}(X), i = 1, …, k, then

∫ G_{V} v dμ = ∑_{i=1}^{k} α_{i} ∫ u ∇_{V_{i}}^{Γ∗}v dμ = ∫ ∑_{i=1}^{k} α_{i}G_{V_{i}} v dμ for all v ∈ FC_{b}^{∞}.

Therefore, we may conclude that

(7.3)  G_{V} = ∑_{i=1}^{k} α_{i}G_{V_{i}}  μ-a.s.

Now let V ⊂ V_{0}(X) be a countable ℚ-vector space such that, for all V ∈ V_{0}(X), there exist V_{n} ∈ V, n ∈ ℕ, such that ∥V − V_{n}∥_{∞} → 0 as n ↑ ∞ and all V_{n} have compact support in a common compact subset of X, where the norm ∥ · ∥_{∞} was defined in (5.5). Such a space V can be easily constructed by using partitions of unity on X. Let Ω_{V} denote the intersection of all sets Ω_{V} with V ∈ V. Then μ(Ω_{V}) = 1. Now take Γ_{0} to be the set of all γ ∈ Ω_{V} such that V ↦ G_{V} (γ) is a ℚ-linear mapping on V. By virtue of (7.3) we thus get μ(Γ_{0}) = 1. Fix γ ∈ Γ_{0}. We have |G_{V} (γ)| ≤ L∥V ∥_{γ}, for all V ∈ V. Hence by the above we can extend this mapping to a linear mapping (again denoted by G_{V} (γ)) defined on the whole of V_{0}(X). Again

|G_{V} (γ)| ≤ L ∥V ∥_{γ} for all V ∈ V_{0}(X).

Hence there is ∇^{Γ}u(γ) ∈ T_{γ}Γ_{X} such that G_{V} (γ) = ⟨∇^{Γ}u(γ), V ⟩_{γ}, and ∥∇^{Γ}u(γ)∥_{γ} ≤ L. Therefore assertion (i) is proved.

The statement (ii) is already settled if V ∈ V. If V ∈ V_{0}(X)\V pick some W in V such that ∥V − W ∥_{∞} ≤ ε, and let (ϕ_{t})_{t∈ℝ} denote the flow generated by W. Then Lemma 5.5 yields

|u(ψ_{t}^{∗}γ) − u(ϕ_{t}^{∗}γ)| ≤ Lρ(ψ_{t}^{∗}γ, ϕ_{t}^{∗}γ) ≤ L t c ε e^{tc} γ(A)^{1/2},

where A = supp V ∪ supp W and c is a constant depending only on V and A. Therefore, if γ ∈ Γ_{0},

|\frac{u(ψ_{t}^{∗}γ) − u(γ)}{t} − ⟨∇^{Γ}u(γ), V ⟩_{γ} |
≤ ε L c e^{tc} γ(A)^{1/2} + ∥∇^{Γ}u(γ)∥_{γ} · ∥V − W ∥_{γ}

+ |\frac{u(ϕ_{t}^{∗}γ) − u(γ)}{t} − ⟨∇^{Γ}u(γ), W ⟩_{γ} |

≤ ε L c e^{tc} γ(A)^{1/2} + ε L γ(A)^{1/2} + o(1)

as t → 0. This proves (ii).

8. Proof of Theorem 1.5 Throughout the proof, take V ⊂ V_{0}(X) as in the proof of Theorem 2.3, let

(ψ_{V,t})_{t∈ℝ} denote the flow of a vector field V ∈ V_{0}(X), and suppose that μ satisfies Assumption 1.1. We will need the following simple lemma.

Lemma 8.1: Suppose (v_{n})_{n∈ℕ} ⊂ L^{2}(μ) converges to 0 in L^{2}(μ) and V ∈ V_{0}(X) is a vector field. Then, for r, t ∈ ℝ, r < t,

∫_{r}^{t} |v_{n} ∘ ψ_{V,s}^{∗}| ds ⟶ 0 in μ-probability as n ↑ ∞.

Proof: Let

Φ̃_{s} := \frac{dμ ⊗ ds}{dμ ∘ (ψ_{V,s}^{∗})^{−1} ⊗ ds},  and  \overline{Φ}_{s} := Φ̃_{s} ∘ ψ_{V,s}^{∗}.

Then (s, γ) ↦ \overline{Φ}_{s}(γ) is jointly measurable, and

\begin{aligned}∫ |v_{n}|^{2} dμ & = \frac{1}{t − r} ∫_{r}^{t} ∫ |v_{n} ∘ ψ_{V,s}^{∗}|^{2}\overline{Φ}_{s} dμ ds \\ & ≥ \frac{1}{t − r} ∫ [ ∫_{r}^{t} |v_{n} ∘ ψ_{V,s}^{∗}| ds]^{2} · [ ∫_{r}^{t} \overline{Φ}_{s}^{−1} ds]^{−1} dμ. \end{aligned}

This implies the assertion, because 0 < ∫_{r}^{t} \overline{Φ}_{s}^{−1} ds < ∞ holds μ-a.s. by 1.1 (d).

Now we will show the first assertion of Theorem 1.5. To this end, suppose that û is ρ-continuous μ-modification of a function u ∈ F with Γ_{E}(u, u)(γ) = ∥∇^{Γ}u(γ)∥_{γ}^{2} ≤ C^{2}, for μ-a.e. γ ∈ Γ_{X}. Choose a sequence (u_{n})_{n∈ℕ} ⊂ 𝔽 converging to u in F. By Lemma 6.2

(8.1)  u_{n}(ψ_{V,t}^{∗}γ) − u_{n}(γ) = ∫_{0}^{t}⟨∇^{Γ}u_{n}, V ⟩_{ψ_{V,s}^{∗}γ} ds,  t ∈ ℝ,

holds for μ-a.e. γ and all V ∈ V_{0}(X) and n ∈ ℕ. Hence there is a measurable subset Ω_{0} of Γ_{X} with μ(Ω_{0}) = 1 such that (8.1) holds for all ω ∈ Ω_{0}, v ∈ V, t ∈ ℝ, and n ∈ ℕ. Next, by applying Lemma 8.1 with v_{n}(γ) := ∥∇^{Γ}u_{n}(γ) − ∇^{Γ}u(γ)∥_{γ}, a diagonalization argument implies the existence of a subsequence (u_{n_{k}})_{k∈ℕ} and a measurable subset Ω_{1} ⊂ Ω_{0} with full μ-measure such that, for all γ ∈ Ω_{1}, V ∈ V, k ∈ ℕ, r < t, and s ∈ ℚ,

\begin{matrix}u_{n_{k}}(ψ_{V,s}^{∗}γ) → û(ψ_{V,s}^{∗}γ),  u_{n_{k}}(γ) ⟶ û(γ),  and \\ ∫_{r}^{t} ∥∇^{Γ}u_{n_{k}} − ∇^{Γ}u∥_{ψ_{V,s}^{∗}γ} ds → 0  as k ↑ ∞. \end{matrix}

Hence

(8.2)  û(ψ_{V,t}^{∗}γ) − û(γ) = ∫_{0}^{t}⟨∇^{Γ}u, V ⟩_{ψ_{V,s}^{∗}γ} ds

is true for all γ ∈ Ω_{1}, t ∈ ℚ, and all V ∈ V. By Lemma 5.2 and the ρ-continuity of û, the identity (8.2) extends to all t ∈ ℝ.

Now let Ω_{2} be the set Ω_{1} ∩ {γ ∈ Γ_{X} | γ satisfies (5.3)}. Then μ(Ω_{2}) = 1. If dim X ≥ 2, ω ∈ Γ_{X}, and γ ∈ Ω_{2} are such that ρ(γ, ω) < ∞, then Proposition 5.4 and Lemma 5.5 yield the existence of a sequence (V_{n})_{n∈ℕ} ⊂ V such that

ρ(ψ_{V_{n},1}^{∗}γ, ω) ≤ \frac{1}{n} and ∫_{0}^{1} ∥V_{n}∥_{ψ_{Vn,t}^{∗}γ}^{2} dt ≤ (ρ(γ, ω) + \frac{1}{n})^{2} for all n ∈ ℕ.

From this, our assumptions, and equation (8.2) we conclude that

(8.3)  |û(γ) − û(ω)| ≤ C lim sup_{n↑∞} \sqrt{∫_{0}^{1} ∥V_{n}∥_{ψ_{Vn,t}^{∗}γ}^{2} dt} = C ρ(γ, ω),

with a similar reasoning if X = ℝ. In particular, û is ρ-Lipschitz continuous on Ω_{2} with Lipschitz constant less or equal to C. Next let

ũ(γ) := sup_{ω∈Ω_{2}} [û(ω) − Cρ(ω, γ)]

if the supremum is finite and ũ(γ) = 0 if not. Then ũ is a μ-measurable function such that ũ = û on Ω_{2} and such that Lip(ũ) ≤ C (cf. McShane (1934)). This proves part (i) of our theorem.

In order to prove ‘≤’ in Theorem 1.5 (ii), fix ω and γ as in the assertion, and consider the function ρ_{ω,r} defined in Lemma 4.2. By Lemma 4.1 (viii), c ∧ ρ_{ω,r} is ρ-Lipschitz continuous with Lip(c ∧ ρ_{ω,r}) ≤ 1, for all c, r > 0. Hence c ∧ ρ_{ω,r} ∈ 𝔽^{(c)} and Γ_{E}(c ∧ ρ_{ω,r}, c ∧ ρ_{ω,r}) ≤ 1 μ-a.e. by Theorem 1.3 and Lemma 4.2 (ii). But c and r were arbitrary, and hence Lemma 4.2 (iii) implies ‘≤’. The inequality ‘≥’ of Theorem 1.5 (ii) follows from the next lemma.

Lemma 8.2: Suppose u ∈ F has a vaguely continuous μ-version û and satisfies Γ_{E}(u, u) ≤ 1 μ-a.s. Then û is already ρ-Lipschitz continuous and Lip(û) ≤ 1

Proof: Since û is ρ-continuous, we already know from (8.3) that, for μ-a.e. ω ∈ Γ_{X}, û has the property that

(8.4)  |û(γ) − û(ω)| ≤ ρ(γ, ω),  for all γ ∈ Γ_{X}.

Now we will show that (8.4) holds for any given ω_{0} ∈ Γ_{X}. So let γ ∈ Γ_{X} be such that ρ(ω_{0}, γ) < ∞, and fix a point in X. Let B_{r} denote the open geodesic ball of radius r around this point, and let ∂B_{r} denote its boundary. Then pick a sequence 0 < r_{1} < r_{2} < ⋯ with r_{k} ↑ ∞ and ω_{0}(∂B_{r_{k}}) = 0. We can write the restriction (ω_{0})_{B_{r_{k}}} of ω_{0} to B_{r_{k}} as ∑_{i=1}^{n_{k}} δ_{y^{i}}. As in the proof of 5.6 we see that (8.5)

U_{k} = {ω ∈ Γ_{X} | ω(∂B_{r_{k}}) = 0 and ω_{B_{r_{k}}} = ∑_{i=1}^{n_{k}} δ_{z^{i}} with ( ∑_{i=1}^{n_{k}} d(z^{i}, y^{i})^{2})^{1/2} < 1/r_{k}}

is open. Now we choose k_{0} ∈ ℕ such that 1/r_{k} < ε and

(8.6)  |û(ω) − û(ω_{0})| < ε  for all ω ∈ U_{k_{0}}.

Note that (8.6) automatically holds for k_{0} replaced by any k ≥ k_{0}, because U_{k} ⊂ U_{k_{0}}. Since μ has full support by Proposition 5.6, there is an ω_{k} ∈ U_{k} such that u satisfies (8.4) for ω_{k} replacing ω. As in (8.5) we write (ω_{k})_{B_{r_{k}}} as ∑_{i=1}^{n_{k}} δ_{z_{k}^{i}}. Since ρ(ω_{0}, γ) < ∞, there exists η^{∗} optimal in the sense of Lemma 4.1 (iv). Pick points x^{1}, …, x^{n_{k}} in the configuration γ such that (y^{1}, x^{1}), …, (y^{n_{k}}, x^{n_{k}}) are points of η^{∗}. Defining γ_{k} := (ω_{k})_{B_{r_{k}}^{C}} + ∑_{i=1}^{n_{k}} δ_{x^{i}}, we get

|û(ω_{0}) − û(γ_{k})| ≤ |û(ω_{0}) − û(ω_{k})| + |û(ω_{k}) − û(γ_{k})|

≤ ε + ρ(ω_{k}, γ_{k})

≤ ε + ( ∑_{i=1}^{n_{k}} d(z_{k}^{i}, x^{i})^{2})^{1/2}

≤ 2ε + ( ∑_{i=1}^{n_{k}} d(y^{i}, x^{i})^{2})^{1/2}

≤ 2ε + ρ(ω_{0}, γ).

Finally observe that ρ(ω_{0}, γ) < ∞ obviously implies that γ_{k} → γ, vaguely as k ↑ ∞.

Acknowledgment: A large part of this work was carried out at the Mathematical Sciences Research Institute in Berkeley, and we would like to thank the institute and the organizers of the Stochastic Analysis year, Ruth Williams and Steven N. Evans, for inviting us. We are also grateful to Byron Schmuland for providing us with the proof of Lemma 2.3 and for his careful reading of a preliminary version of the manuscript.

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Michael Röckner, Fakultät für Mathematik, Universität Bielefeld, D-33615 Bielefeld, Germany

roeckner@mathematik.uni-bielefeld.de

Alexander Schied, Institut für Mathematik, Humboldt-Universität, Unter den Linden 6, D-10099 Berlin, Germany

schied@mathematik.hu-berlin.de