Unitary Brownian motions are linearizable

B. Tsirelson

MSRI, Berkeley, and School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel, e-mail: tsirel@math.tau.ac.il

1991 Mathematical Subject classification: primary 58D20; secondary 22E65, 28C20, 46G12, 46L57, 60B11, 60J65, 81S25.

Brownian motions in the infinite-dimensional group of all unitary operators are studied under strong continuity assumption rather than norm continuity. Every such motion can be described in terms of a countable collection of independent one-dimensional Brownian motions. The proof involves continuous tensor products and continuous quantum measurements. A by-product: a Brownian motion in a separable F-space (not locally convex) is a Gaussian process.

Introduction

The most celebrated and useful random process surely is the standard Brownian motion in ℝ (Wiener process). It is Markovian and Gaussian. Its increments are independent and stationary. Its continuous sample paths are bizarre, but many associated probability distributions are smooth, and connected by wonderful formulas. The multidimensional standard Brownian motion can produce a lot of random processes by means of stochastic differential equations. Especially, it can produce its close relatives, well-known during half a century, — Brownian motions in Lie groups and other topological groups.

A Brownian motion in a Lie group G could be defined constructively, by means of its generator, an invariant differential operator of second order on G, or descriptively, as a continuous G-valued random process with stationary independent increments. The former (constructive) definition stipulates smoothness of the generator; the latter (descriptive) definition does not. Are they equivalent? The question was asked by Seizo Itô, and answered in the positive by Kôsaku Yosida [35]; see also Kiyosi Itô [16], and the book by Henry McKean [24, Sect. 4.7].

Brownian motions in infinite-dimensional groups arise naturally from stochastic differential equations; see [21, Chap. 4] (“temporally homogeneous Brownian flows”), [23, Chap. VIII], [33, Chap. IV], [27, Chap. 6]. Does the constructive approach exhaust all possibilities allowed by the descriptive approach? For the group G = Diff^{r}(M) of all diffeomorphisms of a compact smooth manifold M, the question is answered in the positive by Baxendale [5]; roughly speaking, Brownian motions X in G correspond naturally to Brownian motions Y in the tangent space Δ = T_{e}G to G (at the unit e of G) via the stochastic differential equation (dX) · X^{−1} = dY (in the sense of Stratonovich). For many other groups, for example, the group of all homeomorphisms of a manifold, the very idea of T_{e}G becomes too vague. Is there any constructive approach to Brownian motions in such groups?

Let us split the problem in two. First, we waive the relation Δ = T_{e}G between Δ and G. Instead, we take the Hilbert space Δ = l_{2} once and for all (any other reasonable choice is equivalent, see 1.9). We try to construct a Brownian motion (X, Y) in the direct product G × Δ (the additive group of Δ is meant), whose first component X is the given Brownian motion in G, the second component Y is some Brownian motion in Δ, and the two components are perfectly correlated in the following sense: for each t ∈ (0, ∞) there is a one-to-one correspondence between sample paths of X and Y on [0, t]; that is, the σ-field F_{t}^{X} generated by (X(s))_{s∈[0,t]} coincides with the σ-field F_{t}^{Y} generated by (Y (s))_{s∈[0,t]}. We call such (X, Y) a linearization of X.

So, the first part of the problem is linearizability, that is, existence of a linearization. It is a well-defined question; either X is linearizable, or not. If X is linearizable, we face the second part of the problem: to find a reasonable interpretation of Δ as a kind of T_{e}G, and of the perfect correlation between X and Y as a kind of the stochastic differential equation (dX) · X^{−1} = dY. (It is not a well-defined question, in contrast to linearizability.)

The main result of the present work states that every Brownian motion in the unitary group U(H) of the Hilbert space H (of countable dimension) is linearizable. The linearizability is evidently inherited by each group G that possesses a continuous one-one homomorphism to U(H) (in other words, a faithful unitary representation); for example, the group of all measure preserving transformations of (0, 1), or the group of all diffeomorphisms of a manifold. For the group of all homeomorphisms, however, the question is still open.

It is meant that the group U(H) is equipped with the strong (or, equivalently, weak) operator topology, rather than the norm topology. The distinction may be illustrated by the following simple (commutative) example. Let B_{k} be independent standard Brownian motions in ℝ. Define a (random) unitary operator X(t) by X(t)e_{k} = exp(ic_{k}B_{k}(t))e_{k}, where (e_{k}) is an orthogonal basis (thus, X(t) is diagonal in the basis), and (c_{k}) is a sequence of positive numbers. Then X(·) is norm continuous for c_{k} = 1/\sqrt{log k}, but not for c_{k} = 1. In the strong operator topology, however, X(·) is continuous in any case (even for c_{k} = 2^{k})! Most of well-known results [33], [27] assume norm continuity, or even stronger conditions.

The crucial idea of splitting the constructive approach in two separate problems (linearizability, and its interpretation) was suggested to me by the closely related idea of linearizability in the theory of continuous tensor products of probability spaces (Feldman [13], Tsirelson and Vershik [34]). I am indebted to Anatoly Vershik for drawing my attention to the theory in 1994.

The present paper is self-contained; intersections and parallels with related works are noted, but may be ignored by the reader. Sect. 1 formulates notions and results. Sect. 2 develops a new criterion of linearizability for continuous tensor products of probability spaces. Sect. 3 relates unitary Brownian motions to the theory of continuous quantum measurements (see Davies [11]). Sect. 4 establishes local finiteness of the corresponding quantum stochastic process, which implies linearizability. It is interesting to observe quantum probability helping to classical probability. Naturally, driving forces behind the matter should be more intelligible for readers acquainted with continuous tensor products of Hilbert spaces and probability spaces [2, 13, 3, 4, 34], continuous quantum measurements [11, 22, 7, 1], the relation between the former and the latter [6], and quantum stochastic calculus [25, 26, 15].

Brownian motions in a linear topological space are evidently Gaussian, if the space possesses sufficiently many continuous linear functions. However, L_{p} for 0 < p < 1 is an example of a separable F-space (complete metric vector space) that possesses no non-zero continuous linear functions. It appears that the new criterion of linearizability (Sect. 2) is fulfilled for all Brownian motions in all linear spaces (in fact, in all commutative groups), which is shown in Sect. 5. It bridges a gap between two definitions of Gaussian measures. (Sect. 5 depends on Sect. 2, but does not depend on “quantal” sections 3,4.)

The following definition, used throughout the paper, is borrowed from [17, 5].

Definition. A Brownian motion in a topological group G is a continuous G-valued random process X such that

(a) X(0) = e (the unit of G);

(b) increments on the left,

X(t_{1}), X(t_{2})(X(t_{1}))^{−1}, …, X(t_{n})(X(t_{n−1}))^{−1} ,

are independent whenever 0 ≤ t_{1} ≤ … ≤ t_{n} < ∞; (c) the distribution of X(t)(X(s))^{−1} for 0 ≤ s ≤ t < ∞ depends on t − s only.

For example, any Brownian motion in ℝ (the additive group of ℝ is meant) is of the form X(t) = vt + σB(t) for some v ∈ ℝ, σ ∈ [0, ∞); here B(t) is the standard Brownian motion in ℝ. Any Brownian motion on the circe {z ∈ ℂ : |z| = 1} is of the form X(t) = exp(ivt + iσB(t)). The pair ( exp(ivt + iσB(t)), B(t) ) is a Brownian motion in the product (of the circle and the real line), and forms a linearization of exp(ivt + iσB(t)) (for σ ≠ 0, of course). However, a slightly different terminology is used in the next section: a Brownian motion in a group is treated as a representation of a noise in the group; a linearization of the motion is treated as a linearization of the noise; and a linearization of a noise is treated as a faithful representation of the noise in a linear space.

1. The white noise versus black noises

Multiplication of operators (or composition of transformations) gives rise to oneparametric semigroups of operators (or transformations). Multiplication of measure spaces (or tensor multiplication of Hilbert spaces) should give rise to one-parametric semigroups of such spaces. The idea appeared repeatedly, but still, notions and terminology for spaces are far less standard than these for operators. “Continuous tensor product systems of Hilbert spaces” are defined by Arveson [3] in a framework different from that of a theory of “complete Boolean algebras of type 1 factors” by Araki and Woods [2]. “Factorized Hilbert spaces (and probability spaces) over a Boolean algebra” are defined by Tsirelson and Vershik [34] in a framework different from that of Arveson, and of a theory of “factored probability spaces, indexed by a Borel σ-field” by Feldman [13]. (The short list of approaches is in no way exhaustive.) Throughout the paper I restrict myself to the definition given below, and use the shortest term “noise” borrowed from quantum stochastic calculus.

1.1 Definition. A noise consists of a probability space (Ω, F, P), a one-parametric group (T_{t}) of measure preserving transformations T_{t} : Ω → Ω for t ∈ ℝ, and a twoparametric family (F_{s,t}) of sub-σ-fields F_{s,t} ⊂ F for −∞ < s ≤ t < ∞, such that for all r, s, t

(a) T_{t} sends F_{r,s} onto F_{r+t,s+t} (r ≤ s);

(b) F_{r,s} and F_{s,t} are independent (r ≤ s ≤ t);

(c) F_{r,s} and F_{s,t}, taken together, generate F_{r,t} (r ≤ s ≤ t).

1.2 Note. Here and henceforth, each probability space is assumed to be a Lebesgue space (in the sense of Rokhlin, see [30]), and each σ-field contains all sets of zero probability.

A noise will be called trivial, if each F_{s,t} is trivial, that is, consists of sets of probability 0 or 1 only. Otherwise, each F_{s,t} with s < t is non-atomic.

Remind that a metric (space) is called Polish, if it is complete and separable. A Polish group is a metrizable topological group G that possesses a Polish metric ρ satisfying the condition

ρ(x_{n}, y_{n}) → 0 if and only if both x_{n}y_{n}^{−1} → e and y_{n}^{−1}x_{n} → e

for all x_{n}, y_{n} ∈ G; here e is the unit of G. The condition requires more than ρ to conform to the topology of G. Every metrizable topological group possesses a left-invariant metric ρ_{left} (Birkhoff, Kakutani, see [20]). Existence of a right-invariant metric follows: ρ_{right}(x, y) = ρ_{left}(x^{−1}, y^{−1}). In general, no metric is both left-invariant and right-invariant. Given a leftinvariant metric ρ_{left} and a right-invariant metric ρ_{right}, we may take ρ(x, y) = ρ_{left}(x, y) + ρ_{right}(x, y); we have ρ_{left}(x_{n}, y_{n}) → 0 ⟺ ρ_{left}(y_{n}^{−1}x_{n}, e) → 0 ⟺ y_{n}^{−1}x_{n} → e; similarly, ρ_{right}(x_{n}, y_{n}) → 0 ⟺ ρ_{right}(x_{n}y_{n}^{−1}, e) → 0 ⟺ x_{n}y_{n}^{−1} → e, and we get ρ(x_{n}, y_{n}) → 0 ⟺ (y_{n}^{−1}x_{n} → e and x_{n}y_{n}^{−1} → e). Thus, (x_{n}) is a Cauchy sequence if and only if both x_{m}^{−1}x_{n} → e and x_{n}x_{m}^{−1} → e for m, n → ∞, which does not depend on the choice of ρ_{left}, ρ_{right}. So, G is Polish if and only if it is complete in the metric ρ_{left} + ρ_{right} and separable. See [20], [19, 6-O and 6-Q on pp. 210–213], [5, Sect. 2].

1.3 Definition. Let (Ω, F, P), (T_{t}), (F_{s,t}) form a noise, and G be a Polish group. A representation of the noise in the group is a two-parametric family of G-valued random variables X_{s,t} for −∞ < s ≤ t < ∞, such that for all r, s, t

(a) T_{t} sends X_{r,s} to X_{r+t,s+t} (r ≤ s),

(b) X_{s,t} is measurable w.r.t. F_{s,t} (s ≤ t),
(c) X_{s,t}X_{r,s} = X_{r,t} (s ≤ t),

and for each neighborhoood U of the unit element of the group

(d) P (X_{0,t} ∈ U) → 1 for t → 0+. The representation is called continuous, if for every such U

(e) \frac{1}{t}P (X_{0,t} ∉ U) → 0 for t → 0+. The representation is called faithful, if for each t ≥ 0

(f) the σ-field F_{0,t} is generated by the set {X_{0,s} : s ∈ [0, t]} of random variables.

1.4 Note. A G-valued random variable is an equivalence class of measurable maps Ω → G, the equivalence being equality almost everywhere.

It is well-known [5, Th. 3(i)] that the “continuity condition” (e) is equivalent to continuity of X_{s,t} in s and t for almost all ω ∈ Ω (provided that functions are appropriately chosen within the equivalence classes X_{s,t}).

1.5 Definition. Let G_{1}, G_{2} be Polish groups.

(a) G_{1} is Brown subordinate to G_{2}, if every noise that has a faithful continuous representation in G_{1}, necessarily has a faithful continuous representation in G_{2}.

(b) G_{1} and G_{2} are Brown equivalent, if both G_{1} is Brown subordinate to G_{2}, and G_{2} is Brown subordinate to G_{1}.

The classical result mentioned in Introduction implies that every n-dimensional Lie group is Brown equivalent to the n-dimensional linear space. It is easy to see that an m-dimensional linear space is Brown subordinate to an n-dimensional linear space if and only if m ≤ n. The proof of the following (main) result is finished at the end of Sect. 4.

1.6 Theorem. The infinite-dimensional unitary group U(H) is Brown equivalent to the Hilbert space (that is, to the additive group of the Hilbert space with the norm topology).

1.7 Note. Throughout the paper all Hilbert spaces are assumed to be separable and (unless otherwise stated) of infinite dimension. One Hilbert space, denoted by H, is complex; it is used as a carrier of the unitary group U(H). Another Hilbert space, denoted by Δ, is real; it is used as a carrier of “linear” Brownian motions. The group U(H) is the multiplicative group of all unitary operators on the Hilbert space H. The strong and the weak operator topologies coincide on U(H); this is the topology U(H) is equipped with. It is metrizable. Here is an example of a right-invariant metric: ρ_{right}(U_{1}, U_{2}) = ∑_{k} 2^{−k}∥U_{1}e_{k} − U_{2}e_{k}∥; here e_{1}, e_{2}, … are an orthonormal basis of the Hilbert space. A sequence of unitary operators can converge strongly to a non-invertible isometric operator, which means that ρ_{right} is not complete. However, the metric ρ(U_{1}, U_{2}) = ρ_{right}(U_{1}, U_{2}) + ρ_{right}(U_{1}^{−1}, U_{2}^{−1}) is complete, thus U(H) is a Polish group. (A proof, given in [14, “Weak topology” on pp. 61–64] for the group of all measure preserving transformations of (0, 1), needs only trivial adaptation to U(H).)

1.8 Theorem. Every commutative Polish group is Brown subordinate to the Hilbert space.

A separable F-space may be defined as a linear topological space whose additive group is a Polish group [29, 20]. Local convexity is not assumed. The following facts are evident for locally convex spaces, and well-known for a number of specific F-spaces (such as L_{p} for p < 1). The full generality is achieved now by means of the new approach presented here.

1.9 Corollary. All infinite-dimensional separable F-spaces are Brown equivalent.

1.10 Corollary (preliminary formulation). A Brownian motion in a separable Fspace is a Gaussian process.

See Sect. 5 for the final formulation of 1.10 (given after discussing some definitions of Gaussian processes and measures), and for proofs of 1.8–1.10.

1.11 Conjecture. There is a Polish group not Brown subordinate to the Hilbert space.

If a noise has a faithful continuous representation in some Polish group, then all its representations (in all Polish groups) are necessarily continuous. That is a consequence of Meyer’s theorem on predictability [10] ensuring continuity of all martingales in the corresponding filtration. We may avoid using any theory of martingales by means of an equivalent formulation. Remind that, given a noise and some t > s > 0, the orthogonal projection from L_{2}(Ω, F_{0,t}, P) onto L_{2}(Ω, F_{0,s}, P) is the conditional expectation, f ↦ 𝔼 (f|F_{0,s}). The function 𝔼 (f|F_{0,s}) is defined up to a negligible set depending on s. We avoid the trouble of non-countable union of negligible sets by restricting ourselves to rational numbers s.

1.12 Definition. Let (Ω, F, P), (T_{t}), (F_{s,t}) form a noise. The noise is called predictable, if for any t > 0 and any F_{0,t}-measurable bounded function f : Ω → ℝ, the function 𝔼 (f|F_{0,s}), considered for rational s ∈ (0, t), is uniformly continuous in s for almost all ω ∈ Ω.

The word “predictable”, borrowed from the general theory of processes and filtrations, does not mean that the future of the noise can be predicted from its past. It means rather, that anything is predictable from the infinitesimally near past. Compare it with the Poisson process; its jumps are utterly unexpected, they have no precursors.

1.13 Lemma. If a noise has a faithful continuous representation in some Polish group then the noise is predictable.

1.14 Lemma. If a noise is predictable then all its representations in every Polish group are continuous.

Proofs of 1.13 and 1.14 are left to the reader.

Predictability is defined in a time-asymmetric way, since 𝔼 (f|F_{0,s}) is considered rather than 𝔼 (f|F_{s,t}). (You see, f ≠ 𝔼 (f|F_{0,s}) + 𝔼 (f|F_{s,t}) in general.) Instead of timereverse predictability, we may ask about predictability of time-reverse noise, formed by (Ω′, F′, P′) = (Ω, F, P), T′_{t} = T_{−t}, and F′_{s,t} = F_{−t,−s}. I do not know, whether predictability of a noise implies predictability of the time reverse noise, or not. Also, I do not know, whether every noise is isomorphic to its time reverse, or not. (A similar question is asked by Arveson [3, p. 6] about continuous tensor product systems of Hilbert spaces: “we do not know if an arbitrary product system must be antiisomorphic to itself”.) Anyway, if a noise has a faithful continuous representation in some Polish group then, clearly, the time-reverse noise has a faithful continuous representation in the anti-isomorphic group (the same set G, but ba instead of ab); both noises are predictable, by Lemma 1.13.

If (X_{s,t}) is a representation of a predictable noise in (the additive group of) ℝ, then (X_{0,t})_{t∈[0,∞)} is a continuous process with stationary independent increments, that is, a Brownian motion in ℝ. All such representations are a linear space, that becomes a (real) Hilbert space H_{lin} of finite or countable dimension, being equipped with the norm ∥(X_{s,t})∥ = (𝔼 |X_{0,1}|^{2})^{1/2}. Trivial (non-random) representations X_{s,t} = v(s − t), v ∈ ℝ, form a one-dimensional subspace; its orthogonal complement H_{lin}^{0} consists of centered (that is, zero-mean) representations; if (X_{s,t}) ∈ H_{lin}^{0} and ∥(X_{s,t})∥ = 1, then (X_{0,t})_{t∈[0,∞)} is distributed as the standard Brownian motion (Wiener process). See [3, Sect. 5] for the “dimension of a product system” parallel to our dim(H_{lin}^{0}). We may choose an orthonormal basis (X_{s,t}^{k}), 1 ≤ k < dim H_{lin}, of H_{lin}^{0}. If dim H_{lin}^{0} = d < ∞, we get a representation X_{s,t} = (X_{s,t}^{1}, …, X_{s,t}^{d}) of the noise in ℝ^{d}; otherwise, if dim H_{lin}^{0} = ∞, we may construct a representation X_{s,t} = (c_{1}X_{s,t}^{1}, c_{2}X_{s,t}^{2}, …) in the Hilbert space l_{2}, provided that we choose positive constants c_{1}, c_{2}, … decreasing fast enough. In any case, there is a representation (X_{s,t}) in the Hilbert space, containing (in an evident sense) all representations in ℝ. For any r ≤ t consider the σ-field F_{r,t}^{lin} generated by the set {X_{r,s}|s ∈ [r, t]} of random variables. (The set {X_{s,t}|s ∈ [r, t]} gives the same.) Clearly, (Ω, F, P), (T_{t}), (F_{s,t}^{lin}) form a noise, that may be called the linearizable part of the given predictable noise (Ω, F, P), (T_{t}), (F_{s,t}).

1.15 Definition. A predictable noise is called linearizable, if F_{s,t}^{lin} = F_{s,t} for all s ≤ t.

A predictable noise is linearizable if and only if it has a faithful (continuous) representation in the Hilbert space.

The following fact results from an example of Tsirelson and Vershik [34, Sect. 5].

1.16 Theorem. There is a nontrivial predictable noise with trivial linear part (that is, having only trivial representations in ℝ).

The parallel fact for continuous tensor product systems of Hilbert spaces results from examples of R.T. Powers, see [3, Remark 5.4] and [4, p. 12].

So, a linearizable predictable noise may be decomposed into a finite or countable set of independent copies of the white noise. The opposite extreme is a predictable noise with trivial linear part. Such a noise may be called a black noise. Indeed, terms “white noise” and “coloured noise” (these are not noises as defined here) imply that a noise manifests itself through linear sensors. For a black noise, however, the response of any linear sensor is zero!

What could be a physically reasonable nonlinear sensor able to sense a black noise? Maybe, a fluid could do it, which is hinted at by the following words by Shnirelman [32, p. 3] about a paradoxical motion of an ideal incompressible fluid: “[…] very strong external forces are present, but they are infinitely-fast oscillating in space, and therefore are indistinguishable from zero in the sense of distributions. […] This is the fault of the sensors, not of the forces.”

I do not know, how many nonisomorphic black noises exist, but I believe they are a continuum (accordingly, the section is entitled “the white noise versus black noises”, not “versus the black noise”). An invariant, proposed in the next section, seems to be able to distinguish a continuum of nonisomorphic black noises. A similar question about continuous tensor product systems of Hilbert spaces is asked by Arveson [4, p. 12]: “It is expected that Σ is uncountable, but this has not been proved.” Another question of [4, p. 12] can be answered (in the positive) by means of Theorem 1.16, namely the question, “is there a nontrivial E_{0}-semigroup α with the property that there is a nonzero unit U = {U_{t} : t ≥ 0} and such that every other unit V is related to U by a relation of the form V_{t} = e^{iλt}U_{t}, t ≥ 0, where λ is a complex number?” In our language, a “unit” (for a noise) is a representation in the multiplicative semigroup of complex numbers; all such representations are trivial for a black noise [34, Th. 1.7].

2. Spectral type of a noise

Given a noise (Ω, F, P), (T_{t}), (F_{s,t}), consider the spaces L_{2}(F_{s,t}) = L_{2}(Ω, F_{s,t}, P). If r ≤ s ≤ t then (under the canonical identification) L_{2}(F_{r,s}) ⊗ L_{2}(F_{s,t}) = L_{2}(F_{r,t}), which follows from 1.1(b,c). At the same time L_{2}(F_{r,s}) and L_{2}(F_{s,t}) are linear subspaces of L_{2}(F_{r,t}); note that L_{2}(F_{r,s}) ∩ L_{2}(F_{s,t}) is the one-dimensional space of constants, and L_{2}(F_{r,s}) + L_{2}(F_{s,t}) is (in general) much smaller than L_{2}(F_{r,t}). Introduce σ-fields F_{−∞,t}, F_{t,+∞}, F_{−∞,+∞} naturally (say, F_{−∞,t} is generated by all F_{s,t} with s ∈ (−∞, t]), then F_{s,t} = F_{−∞,t} ∩ F_{s,+∞} and L_{2}(F_{s,t}) = L_{2}(F_{−∞,t}) ∩ L_{2}(F_{s,+∞}).

Denote by E_{s,t} the orthogonal projection from L_{2}(F_{−∞,+∞}) onto L_{2}(F_{s,t}). It is the conditional expectation: E_{s,t}f = 𝔼 (f|F_{s,t}) for −∞ ≤ s ≤ t ≤ +∞. Operators E_{s,t} commute with each other, and E_{s,t} = E_{r,t}E_{s,u} whenever −∞ ≤ r ≤ s ≤ t ≤ u ≤ +∞; in particular, E_{s,t} = E_{−∞,t}E_{s,+∞}.

More generally, given −∞ ≤ s_{1} < t_{1} < … < s_{n} < t_{n} ≤ +∞, we may consider the σ-field generated by σ-fields F_{s_{1},t_{1}}, …, F_{s_{n},t_{n}}; denote it by F_{A} where A = (s_{1}, t_{1}) ∪ … ∪ (s_{n}, t_{n}) is an elementary set, that is, a finite union of intervals. We identify elementary sets that coincide up to a finite number of points; say, (r, s)∪(s, t) is identified with (r, t). More exactly, we define the elementary Boolean algebra A as the factoralgebra of the Boolean algebra generated by intervals, modulo the ideal of finite sets. However, it is usual to say “an elementary set A ∈ A” instead of “an equivalence class A ∈ A”, like “a function f ∈ L_{2}” instead of “an equivalence class f ∈ L_{2}”. For instance, we may say that the complement of (s, t) in A is (−∞, s) ∪ (t, +∞).

So, we have σ-fields F_{A} and Hilbert spaces L_{2}(F_{A}) for A ∈ A, satisfying L_{2}(F_{A∪B}) = L_{2}(F_{A}) ⊗ L_{2}(F_{B}) whenever A ∩ B = ∅ in A. (In terms of [34, Def. 1.2] we have a measure factorization over A.) Corresponding orthogonal projections E_{A} satisfy E_{A}f = 𝔼 (f|F_{A}) and E_{A}E_{B} = E_{B}E_{A} = E_{A∩B}. However, in general E_{A} + E_{B} ≠ E_{A∪B} + E_{A∩B}, that is, (1 − E_{A})(1 − E_{B}) ≠ 1 − E_{A∪B}. The map A ↦ E_{A} is not a homomorphism of Boolean algebras.

Commuting projections E_{A} generate a commutative von Neumann algebra of operators on the Hilbert space L_{2}(F_{−∞,+∞}); the space decomposes into a direct integral of Hilbert spaces over the spectrum of the algebra, see [12, Appendix A84]. The spectrum is a (Lebesgue, see 1.2) measure space (Z, Σ, μ) (though the measure is determined up to equivalence), and each E_{A} becomes (the multiplication by) the indicator function 1_{e(A)} of a set e(A) ∈ Σ (or rather, an equivalence class). The relation E_{A}E_{B} = E_{A∩B} for operators turns into the relation e(A) ∩ e(B) = e(A ∩ B) for sets. So, we have a map, preserving intersections (but not a homomorphism) A ↦ e(A) from the Boolean algebra A to the Boolean algebra Σ mod 0.

2.1 Lemma. For any noise, the σ-field F_{−∞,+∞} is generated by the union of σ-fields F_{(−∞,−ε)∪(ε,+∞)} over all ε > 0.

Proof. In general, an increasing family of σ-fields has at most a countable set of discontinuities (jumps). For the family (F_{−∞,t})_{t∈ℝ} the set of discontinuities must be shiftinvariant due to stationarity, see 1.1(a). Therefore the set is empty; F_{−∞,t} depends on t continuously. In particular, F_{−∞,0} is generated by all F_{−∞,−ε}. Similarly, F_{0,+∞} is generated by all F_{ε,+∞}. However, F_{−∞,0} and F_{0,+∞} together generate F_{−∞,+∞}.

2.2 Corollary. For every t ∈ ℝ

μ(Z \ e((−∞, t − ε) ∪ (t + ε, +∞))) → 0 for ε → 0 .

Proof. E_{(−∞,−ε)∪(ε,+∞)} → 1 for ε → 0 strongly on L_{2}(F_{(−∞,+∞)}) by Lemma 2.1, which proves the case t = 0. The general case follows by stationarity.

Sets e(A) are determined mod 0; to avoid troubles, restrict ourselves to rational elementary sets A (I mean that their boundary points must be rational). For any z ∈ Z consider all rational A such that z ∈ e(A). Such A are a filter (within the “rational” subalgebra), since e(A) ⊂ e(B) whenever A ⊂ B, and e(A ∩ B) = e(A) ∩ e(B). Though, it may happen that z ∈ e(∅), thus “filter” must be understood here as “proper or unproper filter”; the unproper filter contains all A. For each rational t there is A of the filter, bounded away from t, due to Corollary 2.2. Consider the intersection of all A of the filter; boundary points of these A may be included or excluded arbitrarily since, being rational, they cannot belong to the intersection. Denote the intersection by C(z); it is a closed set. For every t, μ{z : t ∈ C(z)} = 0 due to Corollary 2.2. Therefore C(z) is of zero Lebesgue measure (for almost all z). Also, C(z) is bounded, since E_{(−t,t)} → 1 strongly, hence μ(Z \ e((−t, t))) → 0 for t → ∞. So, C(z) is a nowhere dense compact set.

Sets e(A) for rational A ∈ A separate points of Z, therefore z ∈ Z is uniquely determined by the corresponding filter {A : z ∈ e(A)}. The filter, in its turn, is uniquely determined by the corresponding intersection C(z); namely, a rational A ∈ A belongs to the filter if and only if C(z) is contained in the interior of A. So, z ↦ C(z) is an injective map from Z to the set C of all compact subsets of ℝ. The set C becomes a Polish space, being equipped with the Hausdorff metric. The Borel structure corresponding to the metric is the same as the Borel structure generated by sets of the form {C ∈ C : C ⊂ A} for all rational A ∈ A (treated as open sets), therefore the map is measurable. We may identify each z with C(z), Z with C(Z) ⊂ C, μ with a measure on the Polish space C (note that μ(C \ Z) = 0), and Σ with the σ-field of μ-measurable subsets of C, which gives the following result.

2.3 Theorem. Let (Ω, F, P), (T_{t}), (F_{s,t}) form a noise. Then there are: a probability measure μ on the space C (of all compact subsets of ℝ, including the empty set), satisfying the condition

μ{C ∈ C : t ∈ C} = 0 for all t ∈ ℝ ,  (a)

and a direct integral decomposition

L_{2}(Ω, F_{−∞,+∞}, P) = ∫_{C}^{⊕} L̂(C) dμ(C)  (b)

into a measurable field of Hilbert spaces L̂(C) such that for every elementary set (that is, a finite union of intervals) A ⊂ ℝ and every f ∈ L_{2}(Ω, F_{−∞,+∞}, P), decomposed as f = ∫_{C}^{⊕} f̂(C) dμ(C), f̂(C) ∈ L̂(C), the conditional expectation is decomposed as

𝔼 (f|F_{A}) = ∫_{C}^{⊕} f̂(C)1_{e(A)}(C) dμ(C) ,  (c)

where e(A) = {C ∈ C : C ⊂ A}.

The measure μ is determined by the noise up to equivalence.

2.4 Definition. The equivalence class of measures μ on the space C (of all compact subsets of ℝ), appearing in Theorem 2.3, is called the spectral type of the noise. Each such μ is called a spectral measure of the noise.

Given an interval (s, t) ⊂ ℝ, we may use the same construction for decomposing L_{2}(F_{s,t}) into a direct integral of spaces L̂_{s,t}(C) over the space C_{s.t} of all compact subsets of (s, t) (the subsets are bounded away from s and t). Given r < s < t, we get two decompositions of the same Hilbert space,

∫ ^{⊕} L̂_{r,t}(C) dμ_{r,t}(C) = L_{2}(F_{r,t}) = L_{2}(F_{r,s}) ⊗ L_{2}(F_{s,t}) =
= ( ∫ ^{⊕} L̂_{r,s}(C_{1}) dμ_{r,s}(C_{1})) ⊗ ( ∫ ^{⊕} L̂_{s,t}(C_{2}) dμ_{s,t}(C_{2})) =
= ∫ ^{⊕} L̂_{r,s}(C_{1}) ⊗ L̂_{s,t}(C_{2}) dμ_{r,s}(C_{1})dμ_{s,t}(C_{2}) ,

which means that for μ_{r,t}-almost all C

(2.5)  L̂_{r,t}(C) = L̂_{r,s}(C ∩ (r, s)) ⊗ L̂_{s,t}(C ∩ (s, t))

(the case s ∈ C may be neglected), and μ_{r,t} = μ_{r,s} ⊗μ_{s,t}. However, measures μ_{r,s}, μ_{s,t}, μ_{r,t} are not canonical, they are determined up to equivalence; it is better to write

(2.6)  μ_{r,t} ∼ μ_{r,s} ⊗ μ_{s,t} .

(In terms of [34] it is not a measure factorization but a measure type factorization; the distinction is essential, see the example at the end of Sect. 1(c) of [34]. Another example, closer to (2.6), is the random set C = {t : X(t) = 1}, where X is the standard Brownian motion in ℝ; here, C ∩ (r, s) and C ∩ (s, t) are dependent, but their dependence may be expressed by a positive density over the product of marginals.)

On the other hand, L_{2}(F_{r,s}) (as well as L_{2}(F_{s,t})) is a subspace of L_{2}(F_{r,t}) (in terms of the tensor product, f is identified with f ⊗ 1, where 1 is the constant function 1(ω) = 1 treated as a special element of L_{2}(F_{s,t})), and the corresponding orthogonal projection f ↦ 𝔼 (f|F_{r,s}) transforms ∫_{C_{r,t}}^{⊕} f̂(C) dμ_{r,t}(C) into ∫_{C_{r,s}}^{⊕} f̂(C) dμ_{r,s}(C); here C_{r,s} is treated as a subset (rather than a factor) of C_{r,t}. Thus, μ_{r,s} ∼ μ_{r,t}|_{C_{r,s}}; of course, the restricted measure μ_{r,t}|_{C_{r,s}} is defined by (μ_{r,t}|_{C_{r,s}})(E) = μ_{r,t}(E ∩ C_{r,s}). Similarly,

(2.7)  μ_{s,t} ∼ μ|_{C_{s,t}} .

The spectral measure μ on C emerges as follows. Any f ∈ L_{2}(F_{−∞,+∞}) determines a finite measure μ_{f} on C such that

(2.8)  μ_{f}{C ∈ C : C ⊂ A} = ∥𝔼 (f|F_{A})∥^{2}

for all elementary sets A ⊂ ℝ. There is f such that for every g ∈ L_{2}(F_{−∞,+∞}), the corresponding measure μ_{g} is absolutely continuous w.r.t. μ_{f} (in fact, a “generic” f satisfies the condition). For any such f we may take μ = μ_{f}. Of course, μ_{f}(E) = ∫_{E} ∥f̂(C)∥^{2} dμ(C).

In particular, consider the white noise generated by the standard Brownian motion X in ℝ. Any f ∈ L_{2}(F_{−∞,+∞}) decomposes into multiple Itô integrals, f = ∑_{n} ∫…∫ f̂_{n}(t_{1}, …, t_{n}) dX(t_{1}) … dX(t_{n}). For the function 𝔼 (f|F_{A}) the decomposition is the same, but restricted to t_{1}, …, t_{n} belonging to A. The measure μ_{f} is concentrated on finite sets C, and its n-point part is |f̂_{n}(t_{1}, …, t_{n})|^{2} dt_{1} … dt_{n}. So, μ is concentrated on finite sets, and its n-point part may be chosen as the n-dimensional Lebesgue measure. Spaces L̂(C) are one-dimensional, as far as the Brownian motion X is one-dimensional; if it is d-dimensional, then dim L̂({t_{1}, …, t_{n}}) = d^{n}. Note also that the empty set C = ∅ ∈ C is an atom for μ, and L̂(∅) is the one-dimensional space of constants, which holds for any noise. We see that the spectral decomposition of a noise is a generalization of Itô decomposition for the white noise.

Consider the set C_{1} ⊂ C of all single-element sets C, that is, C_{1} = {{t}|t ∈ ℝ}. It may happen that μ(C_{1}) > 0; in that case we get a nontrivial linear subspace

L̂_{1} = ∫_{C_{1}}^{⊕} L̂(C) dμ(C) = ∫ℝ^{⊕} L̂_{1}(t) dt ⊂ L_{2}(F_{−∞,+∞}) ;

the former integral is the same as in Th. 2.3 but restricted to C_{1} ⊂ C; it may be transferred to ℝ by the one-one correspondence C_{1} ∋ C = {t} ↔ t ∈ ℝ, giving the latter integral; the Lebesgue measure (dt) is used, since the transferred measure is shift-invariant up to equivalence. Otherwise (when μ(C_{1}) = 0), L̂_{1} contains only 0. Clearly, f ∈ L̂_{1} if and only if μ_{f}(C \ C_{1}) = 0. Each f ∈ L̂_{1} gives raise to a family (f_{s,t}) of f_{s,t} ∈ L̂_{1} ∩ L_{2}(F_{s,t}) for s ≤ t such that 𝔼 f_{s,t} = 0, and f_{r,s} + f_{s,t} = f_{r,t} whenever r ≤ s ≤ t, and f_{−∞,+∞} = f. (In terms of [34] we have an additive integral, see Def. 1.3 there.) On the other hand, if f ∈ L_{2}(F_{−∞,+∞}) belongs to L_{2}(F_{−∞,t})+L_{2}(F_{t,+∞}) for every t, and 𝔼 f = 0, then f ∈ L̂_{1}. (Proof: μ_{f} is concentrated on {C ∈ C : ∅ ≠ C ⊂ (−∞, t)} ∪ {C ∈ C : ∅ ≠ C ⊂ (t, +∞)} for every t; the intersection over all rational t gives C_{1}.) Remind now the linear part (F_{s,t}^{lin}) of a predictable noise, defined in Sect. 1 (before 1.15).

2.9 Lemma. For every predictable noise, every f ∈ L̂_{1} is measurable w.r.t. F_{−∞,+∞}^{lin}.

Proof. The space L̂_{1} is invariant under the one-parameter unitary group (U_{t}) of time shifts, corresponding to the given group (T_{t}) of measure preserving transformations. Another one-parameter unitary group (V_{λ}) acting on L̂_{1} (but not the whole L_{2}) consists of diagonalizable operators (see [12, Appendix A80]) V_{λ} = ∫_{ℝ}^{⊕} e^{iλt} dt on L̂_{1} = ∫_{ℝ}^{⊕} L̂_{1}(t) dt. The two groups satifsy Weil relation V_{λ}U_{t} = e^{iλt}U_{t}V_{λ}. According to the von Neumann uniqueness theorem (see [28, Th. VIII.14 on p. 275]), L̂_{1} decomposes into direct sum of finite or countable number of irreducible components, — subspaces, each carrying an irreducible representation of (U_{t}), (V_{λ}). Each irreducible representation is unitarily equivalent to the standard representation in L_{2}(ℝ), where U_{t} acts as the shift by t, and V_{λ} acts as the multiplication by e^{iλt}. Comparing the latter with the formula V_{λ} = ∫_{ℝ}^{⊕} e^{iλt} dt we conclude that a function f ∈ L_{2}(ℝ) corresponds to ∫_{ℝ}^{⊕} ĥ(t)f(t) dt for some vector field ĥ(t) ∈ L̂_{1}(t)

(not depending on f). The irreducible component number k (1 ≤ k < 1 + d, where d ∈ {0, 1, …, ∞} is the number of components) determines its vector field ĥ_{k}(t) ∈ L̂_{1}(t), and the set {ĥ_{k}(t)|1 ≤ k < 1 + d} is an orthonormal basis of L̂_{1}(t). Comparing the action of U_{t} on L_{2}(ℝ) and L̂_{1} we conclude that U_{t}ĥ_{k}(s) = ĥ_{k}(s + t). For each k we construct a representation (X_{s,t}^{k}) of the noise in ℝ as follows: X_{s,t}^{k} = ∫_{(s,t)}^{⊕} ĥ_{k}(u) du. All X_{s,t}^{k} are mea-

surable w.r.t. F_{−∞,+∞}^{lin}. Every element of L̂_{1} is of the form ∑_{k} ∫ℝ^{⊕} ĥ_{k}(t)f_{k}(t) dt, therefore it is also measurable w.r.t. F_{−∞,+∞}^{lin}.

The same argument can be applied to a non-predictable noise, giving both Gaussian and Poissonian components of the linear part of the noise, but we do not need it.

Note that dim H_{lin}^{0}, discussed in Sect. 1 (before 1.15), is equal to dim L̂_{1}(t), that is, dim L̂(C) for C ∈ C_{1}.

2.10 Corollary. For every predictable noise, if L̂_{1} generates the whole σ-field F_{−∞,+∞} then the noise is linearizable.

A related result about continuous tensor product systems of Hilbert spaces is given by Arveson [4, Theorem E in Sect. 6]. His “decomposable operators” correspond to a multiplicative counterpart of L̂_{1}, — “multiplicative integrals” in terms of [34], while elements of L̂_{1} are “additive integrals”. The two kinds of integrals generate the same σ-field [34, Th. 1.7].

2.11 Corollary. The following conditions are equivalent for every predictable noise.

(a) The linear part of the noise is trivial.

(b) H_{lin}^{0} = {0}.

(c) L̂_{1} = {0}.

(d) μ(C_{1}) = 0.

(e) μ is concentrated on sets C with no isolated points.

Proof. (a) ⟺(b) by definitions; (b) ⟺(c) since dim H_{lin}^{0} = dim L̂_{1}(t); (c) ⟺(d) by definition of L̂_{1}; (e) ⟹(d) trivially; and (d) ⟹(e) due to (2.7).

Turn to the set C_{finite} ⊂ C of all finite sets C (including the empty set). The corresponding subspace

L̂_{finite} = ∫_{C_{finite}}^{⊕} L̂(C) dμ(C) ⊂ L_{2}(F_{−∞,+∞})

consists of all f ∈ L_{2}(F_{−∞,+∞}) such that μ_{f}(C \ C_{finite}) = 0; the space is non-trivial if and only if μ(C_{finite}) > μ({0}).

2.12 Theorem. L̂_{finite} = L_{2}(F_{−∞,+∞}^{lin}) for every predictable noise.

Proof. L_{2}(F_{−∞,+∞}^{lin}) ⊂ L̂_{finite} due to the decomposition into multiple Ito integrals, since the linearizable part of the noise is generated by independent Brownian motions in ℝ. In order to prove that L̂_{finite} ⊂ L_{2}(F_{−∞,+∞}^{lin}) note that C_{finite} is the union of sets C_{t_{1},...t_{n}} ⊂ C defined for rational t_{1}, …, t_{n} such that −∞ < t_{1} < … < t_{n} < +∞ as follows: C ∈ C_{t_{1},...t_{n}} if and only if each one of the n + 1 intervals (−∞, t_{1}), (t_{1}, t_{2}), …, (t_{n−1}, t_{n}), (t_{n}, +∞) contains no more than one point of C, and no one of the points t_{1}, …, t_{n} belongs to C. Consider the subspace L̂_{t_{1},...,t_{n}} = ∫_{C_{t_{1}...tn}}^{⊕} L̂(C) dμ(C) ⊂ ∫_{C_{finite}}^{⊕} L̂(C) dμ(C) = L̂_{finite}. We have C_{t_{1}...t_{n}} = ({∅} ∪ (−∞, t_{1})) × ({∅} ∪ (t_{1}, t_{2})) × … × ({∅} ∪ (t_{n}, +∞)), where points t are identified with single-point sets {t} ∈ C. Thus, L̂_{t_{1}...t_{n}} = (L̂(∅) ⊕ ∫_{(−∞,t_{1})}^{⊕} L̂_{1}(t) dt) ⊗

(L̂(∅) ⊕ ∫_{(t_{1},t_{2})}^{⊕} L̂_{1}(t) dt) ⊗ … ⊗ (L̂(∅) ⊕ ∫_{(t_{n},+∞)}^{⊕} L̂_{1}(t) dt); here L̂(∅) is the one-dimensional

space of constants. Combining it with Lemma 2.9 we conclude that all elements of L̂_{t_{1},...,t_{n}} are measurable w.r.t. F_{−∞,+∞}^{lin}. It remains to note that the union of all L̂_{t_{1},...,t_{n}} is dense in L̂_{finite}.

2.13 Corollary. If L̂_{finite} generates the whole σ-field F_{−∞,+∞}, then the noise (assumed to be predictable) is linearizable.

2.14 Corollary. The following conditions are equivalent for every predictable noise.

(a) A spectral measure is concentrated on finite sets.

(b) The noise is the product of a finite or countable set of independent copies of the white noise.

2.15 Note. Consider the least α such that a spectral measure is concentrated on sets of Hausdorff dimension ≤ α. It is an invariant of a noise. Probably, the invariant takes on a continuum of values, which could distinguish a continuum of nonisomorphic black noises.

3. From unitary Brownian motions to quantum stochastic processes

Brownian motions in a Lie group G are described by their generators, right-invariant second-order differential operators on G. For an n-dimensional G, such an operator depends on \frac{1}{2}n^{2} + \frac{3}{2}n real parameters. The second-order part of the operator is given by a symmetric tensor (a_{ij}) of diffusion coefficients (at the unit of G), and the first-order part contains n more parameters (b_{i}), often called the drift vector (at the unit of G), though they do not form a vector.

Let G = U(d) be the group of all unitary d × d matrices. The map A ↦ e^{iA} from the linear space Δ of all Hermitian d×d matrices to U(d) is smooth, and smoothly invertible in a neighborhood of the origin, which gives a natural local coordinate system on G; note that n = d^{2}. Denote the inverse map by U ↦ −i log U. A Brownian motion X in G determines a diffusion process (not a Brownian motion) A in Δ, A(t) = −i log X(t), well-defined for small t (until leaving the neighborhood).

The space Δ is a Euclidean space, with the Hilbert-Schmidt scalar product (A, B)_{HS} = tr (AB^{∗}) = tr (AB). Infinitesimal characteristics (b_{i}), (a_{ij}) of X may be identified with the following linear and bilinear forms on Δ: for all B, C ∈ Δ,

b(B) = \frac{d}{dt}|_{t=0}𝔼 tr (A(t)B) , a(B, C) = \frac{d}{dt}|_{t=0}𝔼 (tr (A(t)B) · tr (A(t)C)) .

There is another interesting parametrization. Matrices x_{t} = 𝔼 X(t) form a oneparametric semigroup, therefore x_{t} = e^{tY} for some Y; the matrix Y represents the drift of X in the linear space of all matrices (while b represents the drift of X in the manifold of unitary matrices). In order to get parameters describing the spread of X, note that each U ∈ U(d) determines an automorphism M ↦ UMU^{∗} of the matrix algebra M_{d}(ℂ), and the automorphism is a quadratic function of U. We define T_{t}(M) = 𝔼 (X(t)MX^{∗}(t)), which gives a one-parameter semigroup of linear maps T_{t} : M_{d}(ℂ) → M_{d}(ℂ); thus, T_{t} = e^{tZ} for some linear map Z : M_{d}(ℂ) → M_{d}(ℂ). (Do not confuse T_{t} with the time shift T_{t} introduced by Def. 1.1.) The following lemma shows that the pair (Y, Z) determines a and b uniquely. However, the result will not be used, and its proof is relegated to Appendix.

3.1 Lemma. (a) The semigroup (T_{t}) determines uniquely the law of the Brownian motion (det X(t))^{−1/d}X(t) in the group SU(d).

(b) The two semigroups (x_{t}), (T_{t}) determine uniquely the law of the Brownian motion X(t) in the group U(d).

Turn to the infinite-dimensional case: G = U(H) is the group of all unitary operators in the Hilbert space (see 1.7). We do not know, how to generalize infinitesimal characteristics a and b for all Brownian motions X in G (some cases are investigated in [33,27]). In contrast, Y and Z have a straightforward generalization. Still, operators

(3.2)  x_{t} = 𝔼 X(t)

are well-defined, ∥x_{t}∥ ≤ 1, and form a strongly continuous semigroup. The general theory of operator semigroups ensures that the semigroup has its generator Y, densely defined, usually unbounded (for detail, see [11]). However, we do not need the generator; what we need is the very semigroup (x_{t}). Another semigroup (T_{t}) is formed by bounded linear operators T_{t} : V → V, where V is the Banach space of all Hermitian trace-class operators on the Hilbert space H (see [11]); note that H is complex, while V is real. Operators T_{t} are defined as before:

(3.3)  T_{t}(ρ) = 𝔼 (X(t)ρX^{∗}(t))

for ρ ∈ V; they form a strongly continuous semigroup, and ∥T_{t}∥ ≤ 1. In fact, (T_{t}) is a special case of a so-called quantum dynamical semigroup (see [22]), and X(·) is one of socalled enravelings of (T_{t}) (see [7]; physicists are more interested in nonlinear enravelings). So, any Brownian motion X in U(H) determines two semigroups, (x_{t}) on H and (T_{t}) on V. I do not know, whether X is uniquely determined by the semigroups, or not.

Remind the spectral measure μ on C, and μ_{s,t} on C_{s,t}. Denote by Σ the σ-field of all μ-measurable subsets of C, and by Σ_{s,t} the σ-field of all μ_{s,t}-measurable subsets of C_{s,t}.

3.4 Theorem. For every Brownian motion in the unitary group U(H), there is one and only one family (E_{s,t})_{−∞<s<t<+∞} satisfying the following conditions (a)–(f).

(a) For any s < t, E_{s,t} is a map, defined on the σ-field Σ_{s,t}, taking on values in the algebra of all bounded linear maps from V to itself, and E_{s,t}(E_{1}) = E_{s,t}(E_{2}) whenever E_{1}, E_{2} differ by a μ_{s,t}-negligible set.

(b) The map E_{s,t} is an instrument, as defined in [11, Chap. 4, Def. 1.1]. It means, first, countable additivity: for any sequence E_{1}, E_{2}, … of disjoint sets in Σ_{s,t}

E_{s,t}(E_{1} ∪ E_{2} ∪ …)ρ = E_{s,t}(E_{1})ρ + E_{s,t}(E_{2})ρ + …  (b1)

where the sum is norm convergent for each ρ ∈ V; second, positivity:

E_{s,t}(E)ρ ∈ V_{+} for all ρ ∈ V_{+}, E ∈ Σ_{s,t};  (b2)

(here V_{+} = {ρ ∈ V : ∀h ∈ H (ρh, h) ≥ 0}); and third, trace conservation:

tr (E_{s,t}(C_{s,t})ρ) = tr (ρ) for all ρ ∈ V .  (b3)

(There is one more condition, complete positivity, see [11, Sect. 9.2]; in fact, it is also satisfied, which, however, will be neither proved nor used.)

(c) E_{r,t}(E_{1} × E_{2}) = E_{s,t}(E_{2})E_{r,s}(E_{1}) whenever r < s < t, E_{1} ∈ Σ_{r,s}, E_{2} ∈ Σ_{s,t}; here E_{1} × E_{2} means {C ∈ C_{r,t} : C ∩ (r, s) ∈ E_{1} and C ∩ (s, t) ∈ E_{2}}.

(d) E_{r+t,s+t}(T_{t}^{−1}E) = E_{r,s}(E) whenever r < s, t ∈ ℝ, E ∈ Σ_{r,s}.

(e) E_{0,t}(C_{0,t})ρ → ρ (in norm) when t → 0+, for every ρ ∈ V.

(f) E_{s,t}(C_{s,t}) = T_{t−s}, and E_{s,t}({∅})ρ = x_{t−s}ρx_{t−s}^{∗} whenever ρ ∈ V and −∞ < s < t < +∞. (Here (x_{t}) and (T_{t}) are defined by (3.2), (3.3).)

Before starting the proof, remind of some well-known notions and facts. Let ψ ∈ G⊗H, where G is another Hilbert space. For any h ∈ H define (ψ, h)_{H} ∈ G by the equality ((ψ, h)_{H}, g) = (ψ, g ⊗ h) for all g ∈ G. For any bounded linear operator U : H → G ⊗ H define a bounded linear operator T_{U} : V → V as follows. Choose an orthonormal basis (g_{k}) in G, and define A_{1}, A_{2}, …: H → H by Uh = ∑_{k} g_{k} ⊗ A_{k}h for all h ∈ H, then T_{U}(ρ) = ∑_{k} A_{k}ρA_{k}^{∗}. For the one-dimensional operator ρ_{h} ∈ V defined by ρ_{h}x = (x, h)h for all x ∈ H we have (T_{U}(ρ_{h})h_{2}, h_{1}) = ∑_{k}(A_{k}h, h_{1})\overline{(A_{k}h, h_{2})} = ((Uh, h_{1})_{H}, (Uh, h_{2})_{H}) for all h_{1}, h_{2} ∈ H, which shows that T_{U} does not depend on the choice of the basis (g_{k}), since linear combinations of ρ_{h} are dense in V.

Let F, G be Hilbert spaces, and U_{1} : H → F ⊗ H, U_{2} : H → G ⊗ H. Define U : H → F ⊗ G ⊗ H as U = (1_{F} ⊗ U_{2})U_{1}. If U_{1}h = ∑_{k} f_{k} ⊗ A_{k}h and U_{2}h = ∑_{l} g_{l} ⊗ B_{l}h, then Uh = ∑_{k} f_{k} ⊗ U_{2}A_{k}h = ∑_{k,l} f_{k} ⊗ g_{l} ⊗ B_{l}A_{k}h. It is easy to see that T_{U} = T_{U_{2}}T_{U_{1}}, that is, T_{U}(ρ) = T_{U_{2}}(T_{U_{1}}(ρ)) for all ρ ∈ V.

Let U : H → G ⊗ H be an isometric operator, and G be a direct integral over some measure space, G = ∫_{Λ}^{⊕} G(λ) dμ(λ). For any μ-measurable set E ⊂ Λ consider the projection Q_{E} = ∫_{Λ}^{⊕} 1_{E}(λ) dμ(λ) on G; that is, if g = ∫_{Λ}^{⊕} g(λ) dμ(λ), then Q_{E}g = ∫_{E}^{⊕} g(λ) dμ(λ). Define E(E) = T_{(Q_{E}⊗1_{H})U}. Then E is an instrument.

Proof of Theorem 3.4. Uniqueness: (c) and (f) determine E_{s,t}(e_{s,t}(A)), where e_{s,t}(A) = {C ∈ C_{s,t} : C ⊂ A} and A ⊂ ℝ is a finite union of intervals. Sets e_{s,t}(A) generate Σ_{s,t} (mod 0), and e_{s,t}(A ∩ B) = e_{s,t}(A) ∩ e_{s,t}(B). Therefore, E_{s,t} is determined on the whole Σ_{s,t}.

Existence. Introduce an isometric operator X̃_{t} : H → L_{2}(F_{0,t}, H) by (X̃_{t}h)(ω) = X(t, ω)h for h ∈ H, ω ∈ Ω. Identifying L_{2}(F_{0,t}, H) with L_{2}(F_{0,t}) ⊗ H = (∫_{C_{0,t}}^{⊕} L̂_{0,t}(C) dμ_{0,t}(C)) ⊗ H we have an isometric operator X̃_{t} : H → (∫^{⊕} …) ⊗ H. An instrument E_{0,t} appears, E_{0,t}(E) = T_{(Q_{E}⊗1_{H})X̃_{t}}. The same construction for X_{s,t} = X(t)(X(s))^{−1} instead of X_{0,t} = X(t) gives E_{s,t}. The construction is invariant under the time shift group (T_{t}). Thus, (a), (b), and (d) hold.

For any h_{1}, h_{2} ∈ H we have (X̃_{t}h_{2}, h_{1})_{H} = (X(t)h_{2}, h_{1}), since (X̃_{t}h_{2}, g ⊗ h_{1}) = ∫_{Ω}(X(t, ω)h_{2}, g(ω)h_{1}) dP (ω) = ∫_{Ω}(X(t, ω)h_{2}, h_{1})g(ω) dP (ω) for all g ∈ L_{2}(F_{0,t}).

For any ψ ∈ L_{2}(F_{0,t}) ⊗ H, h ∈ H, and E ∈ Σ_{0,t} we have ((Q_{E} ⊗ 1_{H})ψ, h)_{H} = Q_{E}(ψ, h)_{H}, since the two expressions are linear in ψ, and coincide on factorizable vectors; indeed, if ψ = f ⊗ h_{1}, f ∈ L_{2}(F_{0,t}), h_{1} ∈ H, then both expressions turn into (h_{1}, h)Q_{E}f.

Take ψ = X̃_{t}h, then (ψ, h_{1})_{H} = (X(t)h, h_{1}), thus, ((Q_{E} ⊗ 1_{H})X̃_{t}h, h_{1})_{H} = Q_{E}(X(t)h, h_{1}). Applying the formula (T_{U}(ρ_{h})h_{2}, h_{1}) = ((Uh, h_{1})_{H}, (Uh, h_{2})_{H}) for U = (Q_{E} ⊗ 1_{H})X̃_{t}, we get the matrix element

(3.5)  (E_{0,t}(E)(ρ_{h})h_{2}, h_{1}) = (Q_{E}(X(t)h, h_{1}), Q_{E}(X(t)h, h_{2}))

for all h, h_{1}, h_{2} ∈ H, E ∈ Σ_{0,t}.

For proving (f), consider the two extreme cases, E = {∅} and E = C_{0,t}. For E = C_{0,t} we have Q_{E} = 1_{L_{2}(F_{0,t})} and (E_{0,t}(C_{0,t})(ρ_{h})h_{2}, h_{1}) = ((X(t)h, h_{1}), (X(t)h, h_{2})) = 𝔼 (X(t)h, h_{1})\overline{(X(t)h, h_{2})}. On the other hand, (X(t)ρ_{h}X^{∗}(t)h_{2}, h_{1}) = (X^{∗}(t)h_{2}, h)(X(t)h, h_{1}) = (X(t)h, h_{1})\overline{(X(t)h, h_{2})}, and we get (E_{0,t}(C_{0,t})(ρ_{h})h_{2}, h_{1}) = 𝔼 (X(t)ρ_{h}X^{∗}(t)h_{2}, h_{1}) = (T_{t}(ρ_{h})h_{2}, h_{1}) by (3.3). It means that E_{0,t}(C_{0,t}) = T_{t}. Similarly (or by (e)), E_{s,t}(C_{s,t}) = T_{t−s}, which is the first claim of (f). For the other case, E = {∅}, we have (Q_{{∅}}f)(ω) = 𝔼 f for all ω ∈ Ω, f ∈ L_{2}(F_{0,t}), that is, Q_{{∅}}f = (𝔼 f)1_{Ω}, which is 2.3(c) for A = (−∞, 0)∪(t, +∞). Thus, (E_{0,t}({∅})(ρ_{h})h_{2}, h_{1}) = (1_{Ω}𝔼 (X(t)h, h_{1}), 1_{Ω}𝔼 (X(t)h, h_{2})) = 𝔼 (X(t)h, h_{1})·\overline{𝔼 (X(t)h, h_{2})} = (x_{t}h, h_{1})\overline{(x_{t}h, h_{2})}. On the other hand, (x_{t}ρ_{h}x_{t}^{∗}h_{2}, h_{1}) = (x_{t}^{∗}h_{2}, h)(x_{t}h, h_{1}) = (x_{t}h, h_{1})\overline{(x_{t}h, h_{2})}, and we get E_{0,t}({∅})(ρ_{h}) = x_{t}ρ_{h}x_{t}^{∗} for all h ∈ H. Therefore E_{0,t}({∅})(ρ) = x_{t}ρx_{t}^{∗} for all ρ ∈ V. Similarly (or by (d)), E_{s,t}({∅})ρ = x_{t−s}ρx_{t−s}^{∗}, which completes the proof of (f).

Item (e) is reduced by (f) to strong continuity of the semigroup (T_{t}), that is, T_{t}ρ → ρ when t → 0. It suffices to prove it for ρ = ρ_{h}, h ∈ H. We have T_{t}(ρ_{h}) = 𝔼 (X(t)ρ_{h}X^{∗}(t)) = 𝔼 ρ_{X(t)h}; however, ∥ρ_{X(t)h} − ρ_{h}∥ ≤ 2∥X(t)h − h∥ → 0 when t → 0 for almost all ω, and never exceeds 2∥h∥; therefore ∥T_{t}(ρ_{h}) − ρ_{h}∥ ≤ 2𝔼 ∥X(t)h − h∥ → 0, which proves (e).

Before proving (c), note that for any r < s < t, any linear operator U : H → L_{2}(F_{s,t}, H) and any vector-function ψ ∈ L_{2}(F_{r,s}, H), the vector-function ξ = (1_{L_{2}(F_{r,s})} ⊗ U)ψ is given by ξ(ω) = (U(ψ(ω)))(ω). Indeed, it suffices to consider a factorizable ψ = f ⊗h, that is, ψ(ω) = f(ω)h, f ∈ L_{2}(F_{r,s}), h ∈ H. Then ξ = f ⊗Uh, that is, ξ(ω) = f(ω)· ((Uh)(ω)), while U(ψ(ω)) = U(f(ω)h) = f(ω)·(Uh) and (U(ψ(ω)))(ω) = f(ω)·((Uh)(ω)), too.

Now assume that r < s < t, E_{1} ∈ Σ_{r,s}, E_{2} ∈ Σ_{s,t}; we have to prove that E_{r,t}(E_{1}×E_{2}) = E_{s,t}(E_{2})E_{r,s}(E_{1}). Similarly to X̃_{0,t}, introduce X̃_{s,t} : H → L_{2}(F_{s,t}, H) = L_{2}(F_{s,t}) ⊗ H by (X̃_{s,t}h)(ω) = X_{s,t}(ω)h; the same for X̃_{r,s} and X̃_{r,t}. Denote F = L_{2}(F_{r,s}), G = L_{2}(F_{s,t}), U_{1} = (Q_{E_{1}} ⊗ 1_{H})X̃_{r,s} : H → F ⊗ H, U_{2} = (Q_{E_{2}} ⊗ 1_{H})X̃_{s,t} : H → G ⊗ H, and U = (Q_{E_{1}×E_{2}} ⊗ 1_{H})X̃_{r,t} : H → F ⊗ G ⊗ H, then (c) takes the form T_{U} = T_{U_{2}}T_{U_{1}}. It suffices to check that U = (1_{F} ⊗ U_{2})U_{1}. A simpler equality X̃_{r,t} = (1_{F} ⊗ X̃_{s,t})X̃_{r,s} holds, since ((1_{F} ⊗ X̃_{s,t})X̃_{r,s}h)(ω) = (X̃_{s,t}((X̃_{r,s}h)(ω)))(ω) = (X̃_{s,t}(X_{r,s}(ω)h))(ω) = X_{s,t}(ω)X_{r,s}(ω)h = X_{r,t}(ω)h = (X̃_{r,t}h)(ω). Also, Q_{E_{1}×E_{2}} = Q_{E_{1}} ⊗Q_{E_{2}}. Thus, U = (Q_{E_{1}} ⊗ Q_{E_{2}} ⊗1_{H})(1_{F} ⊗X̃_{s,t})X̃_{r,s}, while (1_{F} ⊗U_{2})U_{1} = (1_{F} ⊗(Q_{E_{2}} ⊗1_{H})X̃_{s,t})(Q_{E_{1}} ⊗1_{H})X̃_{r,s}. It remains to check that (Q_{E_{1}} ⊗ Q_{E_{2}} ⊗ 1_{H})(1_{F} ⊗ X̃_{s,t}) = (1_{F} ⊗ (Q_{E_{2}} ⊗ 1_{H})X̃_{s,t})(Q_{E_{1}} ⊗ 1_{H}). The two linear operators F ⊗ H → F ⊗ G ⊗ H coincide on the whole F ⊗ H, since they coincide on factorizable vectors; for any f ∈ F, h ∈ H both operators transform f ⊗ h into Q_{E_{1}}f ⊗ (Q_{E_{2}} ⊗ 1_{H})X̃_{s,t}h. So, (c) is verified.

The measure μ_{f} on C_{0,t}, defined by (2.8) for any f ∈ L_{2}(F_{0,t}), may be written in terms of spectral projections Q_{E} as μ_{f}(E) = (Q_{E}f, f) = ∥Q_{E}f∥^{2}. Let f be a matrix element of X(t), that is, f = (X(t)h_{2}, h_{1}) for arbitrary h_{1}, h_{2} ∈ H, t ∈ [0, ∞). There is a simple relation between the scalar-valued measure μ_{f} and the operator-valued measure E_{0,t}. The former is a matrix element of the latter:

(3.6)  μ_{f}(E) = (E_{0,t}(E)ρ_{h_{2}}, ρ_{h_{1}})_{HS} for f = (X(t)h_{2}, h_{1});

here (·, ·)_{HS} is the scalar product in the Hilbert space of all Hilbert-Schmidt operators in H, that is, (ρ_{1}, ρ_{2})_{HS} = tr (ρ_{1}ρ_{2}) for ρ_{1}, ρ_{2} ∈ V. (Only trace-class Hermitian operators are really used here.) Taking into account that (A, ρ_{h})_{HS} = (Ah, h) for every A, we can deduce (3.6) from (3.5) as follows: (E_{0,t}(E)ρ_{h_{2}}, ρ_{h_{1}})_{HS} = (E_{0,t}(E)(ρ_{h_{2}})h_{1}, h_{1}) = (Q_{E}(X(t)h_{2}, h_{1}), Q_{E}(X(t)h_{2}, h_{1})) = ∥Q_{E}(X(t)h_{2}, h_{1})∥^{2} = ∥Q_{E}f∥^{2} = μ_{f}(E).

Conditions (a)–(e) of Theorem 3.4 define a notion close to the notion of a quantum stochastic process as defined in [11, Sect. 5.2]. There are two distinctions. First, Davies stipulates “the value space X”; assuming that his X is a single point, we get rid of it. Second, Davies considers only finite sets C, rather than all compact sets. This is the point! It will be shown in the next section, that our operator-valued measure E_{s,t} is concentrated on C_{finite} ∩ C_{s,t}. Thus, the noise will appear to be linearizable, and (E_{s,t}) will appear to be a quantum stochastic process in the sense of Davies. However, his “bounded interaction rate” condition [11, (2.9) in Sect. 5.2] does not hold in our framework; finiteness of sets C is ensured by a more subtle mechanism described in the next section. Note also that the quantum dynamical semigroup (T_{t}) is strongly continuous (T_{t}ρ → ρ) but in general not norm continuous; compare it with the following phrases of Lindblad: “[…] we have to assume that the semigroup is norm continuous […] a condition which is not fulfilled in many applications. (We may hope that this restriction can be ultimately removed using more powerful mathematics.)” [22, p. 120].

4. A compactness argument

Let X be a Brownian motion in the unitary group U(H), and (E_{s,t}) the corresponding family of instruments, given by Theorem 3.4. Let ρ ∈ V_{1}^{+}, that is, ρ : H → H, ρ ≥ 0, and tr (ρ) = 1. A probability measure μ_{s,t}(ρ, ·) on (C_{s,t}, Σ_{s,t}) arises as follows: μ_{s,t}(ρ, E) = tr (E_{s,t}(E)ρ) for E ∈ Σ_{s,t}. The measure depends linearly on the parameter ρ. The V -valued measure E ↦ E_{s,t}(E)ρ has its V -valued density p_{s,t}(ρ, ·) w.r.t. μ_{s,t}(ρ, ·); it is a μ_{s,t}(ρ, ·)- measurable function C_{s,t} ∋ C ↦ p_{s,t}(ρ, C) ∈ V_{1}^{+} (determined uniquely μ_{s,t}(ρ, ·)-almost everywhere) such that E_{s,t}(E)ρ = ∫_{E} p_{s,t}(ρ, C) μ_{s,t}(ρ, dC) for all E ∈ Σ_{s,t}. Its existence can be checked easily by considering scalar-valued measures E ↦ ((E_{s,t}(E)ρ)h_{2}, h_{1}) for h_{1}, h_{2} ∈ H, and their densities (Radon-Nikodym derivatives) w.r.t. μ_{s,t}(ρ, ·).

Property 3.4(c), E_{r,t}(E_{1} × E_{2}) = E_{s,t}(E_{2})E_{r,s}(E_{1}), may be reformulated in terms of p and μ as follows. Let ρ_{0} ∈ V_{1}^{+}. Consider ρ_{1} = E_{r,s}(E_{1})ρ_{0} = ∫_{E_{1}} p_{r,s}(ρ_{0}, C_{1})μ_{r,s}(ρ_{0}, dC_{1}). We have ∫_{E_{1}×E_{2}} p_{r,t}(ρ_{0}, C)μ_{r,t}(ρ_{0}, dC) = E_{r,t}(E_{1} × E_{2})ρ_{0} = E_{s,t}(E_{2})ρ_{1} = ∫_{E_{1}} E_{s,t}(E_{2})p_{r,s}(ρ_{0}, C_{1})μ_{r,s}(ρ_{0}, dC_{1}) = ∫_{E_{1}} μ_{r,s}(ρ_{0}, dC_{1}) ∫_{E_{2}} μ_{s,t}(p_{r,s}(ρ_{0}, C_{1}), dC_{2})p_{s,t}(p_{r,s}(ρ_{0}, C_{1}), C_{2}) for all E_{1} ∈ Σ_{r,s}, E_{2} ∈ Σ_{s,t}. Taking the trace we get

(4.1)  μ_{r,t}(ρ_{0}, E_{1} × E_{2}) = ∫_{E_{1}} μ_{r,s}(ρ_{0}, dC_{1})μ_{s,t}(p_{r,s}(ρ_{0}, C_{1}), E_{2})

for E_{1} ∈ Σ_{r,s}, E_{2} ∈ Σ_{s,t}, and then

(4.2)  p_{r,t}(ρ_{0}, C_{1} ∪ C_{2}) = p_{s,t}(p_{r,s}(ρ_{0}, C_{1}), C_{2})

for almost all C_{1} ∈ C_{r,s}, C_{2} ∈ C_{s,t}. The property (4.1) can be generalized to arbitrary E ∈ Σ_{r,t}; denote E(C_{1}) = {C_{2} ∈ C_{s,t} : C_{1} ∪ C_{2} ∈ E} for C_{1} ∈ C_{r,s}, then

(4.3)  μ_{r,t}(ρ_{0}, E) = ∫_{C_{r,s}} μ_{r,s}(ρ_{0}, dC_{1})μ_{s,t}(p_{r,s}(ρ_{0}, C_{1}), E(C_{1})) .

It holds by (4.1) for every product set E = E_{1} × E_{2}, by additivity for finite unions of product sets, and by continuity for all sets E. Our next step is to make explicit a Markov property implicit in (4.2), (4.3).

Introduce Borel spaces Y_{t} = C_{−∞,t} ⊗V_{1}^{+}. Let s < t. The instrument E_{s,t} gives raise to a transition probability P_{s,t}, that is, a Borel function on Y_{s} whose values are probability measures on Y_{t}. Before giving a formal definition, consider the idea. We have y_{0} ∈ Y_{s}, that is, y_{0} = (C_{0}, ρ_{0}), C_{0} ∈ C_{−∞,s} and ρ_{0} ∈ V_{1}^{+}. The latter determines the corresponding probability distribution μ_{s,t}(ρ_{0}, ·) for a random element C of C_{s.t}. Choose C at random; calculate the corresponding ρ_{1} = p_{s,t}(ρ_{0}, C) ∈ V_{1}^{+}. We get a (random) y_{1} ∈ Y_{t}, namely, y_{1} = (C_{1}, ρ_{1}), where C_{1} = C_{0} ∪ C. The distribution of y_{1} is the measure P_{s,t}(y_{0}) on Y_{t} that corresponds to y_{0} ∈ Y_{s}. Formally,

(4.4)  P_{s,t}((C_{0}, ρ_{0}), A) = μ_{s,t}(ρ_{0}, {C ∈ C_{s,t} : (C_{0} ∪ C, p_{s,t}(ρ_{0}, C)) ∈ A})

for all C_{0} ∈ C_{−∞,s}, ρ_{0} ∈ V_{1}^{+}, and Borel sets A ⊂ Y_{t}. The Markov property is

(4.5)  P_{r,t}(y_{0}, A) = ∫_{Y_{s}} P_{r,s}(y_{0}, dy_{1})P_{s,t}(y_{1}, A)

for all y_{0} ∈ Y_{r} and all Borel sets A ⊂ Y_{t}. A proof follows. We have y_{0} = (C_{0}, ρ_{0}), C_{0} ∈ C_{−∞,r}, ρ_{0} ∈ V_{1}^{+}. Introduce E = {C ∈ C_{r,t} : (C_{0} ∪ C, p_{r,t}(ρ_{0}, C)) ∈ A}, then P_{r,t}(y_{0}, A) = μ_{r,t}(ρ_{0}, E) by (4.4), and μ_{r,t}(ρ_{0}, E) is given by (4.3). The right-hand side of (4.5) is an integral over y_{1} ∈ Y_{s}, but all relevant y_{1} are of the form y_{1} = (C_{0} ∪ C_{1}, p_{r,s}(ρ_{0}, C_{1})), and the distribution P_{r,s}(y_{0}, ·) for y_{1} results from the distribution μ_{r,s}(ρ_{0}, ·) for C_{1} by (4.4). Now (4.5) takes the form ∫_{C_{r,s}} μ_{r,s}(ρ_{0}, dC_{1})μ_{s,t}(p_{r,s}(ρ_{0}, C_{1}), E(C_{1})) = ∫_{C_{r,s}} μ_{r,s}(ρ_{0}, dC_{1})P_{s,t}((C_{0} ∪ C_{1}, p_{r,s}(ρ_{0}, C_{1})), A); it remains to note that P_{s,t}((C_{0} ∪ C_{1}, p_{r,s}(ρ_{0}, C_{1})), A) = μ_{s,t}(p_{r,s}(ρ_{0}, C_{1}), E(C_{1})) for C_{1} ⊂ C_{r,s} by (4.4) since, using (4.2), {C_{2} ∈ C_{s,t} : (C_{0} ∪ C_{1} ∪ C_{2}, p_{s,t}(p_{r,s}(ρ_{0}, C_{1}), C_{2})) ∈ A} = {C_{2} ∈ C_{s,t} : (C_{0} ∪ C_{1} ∪ C_{2}, p_{r,t}(ρ_{0}, C_{1} ∪ C_{2})) ∈ A} = {C_{2} ∈ C_{s,t} : C_{1} ∪ C_{2} ∈ E} = E(C_{1}).

The Markov property (4.5) allows us to introduce a Markov process (Y_{t}). However, having (for now) no information on its regularity, we restrict ourselves to rational values of t, thus avoiding the choice of a modification. Strong Markov property is not claimed, only the simple (fixed-time) Markov property. Given ρ_{0} ∈ V_{1}^{+}, there is a Markov process (Y_{t}), defined for all rational t ∈ [0, ∞), such that Y_{t} takes on values in Y_{t} = C_{−∞,t} × V_{1}^{+}, that is,

(4.6) Y_{t} = (C_{t}, ρ_{t}), Y_{0} = (∅, ρ_{0}) with probability 1, ℙ (Y_{t} ∈ A|Y_{s}) = P_{s,t}(Y_{s}, A) for s < t, A ⊂ Y_{t}.

The specific form (4.4) of transition probabilities implies the following. First, C_{t} ∩ (−∞, 0) = ∅, and C_{t} ∩ (0, s) does not depend on t as far as t ≥ s ≥ 0. Second, ρ_{t} = p_{s,t}(ρ_{s}, C_{t} ∩ (s, t)) for t > s ≥ 0 (it is meant that s, t are rational). Third, ℙ (C_{t} ∩ (s, t) ∈ E|C_{s}, ρ_{s}) = μ_{s,t}(ρ_{s}, E) = tr (E_{s,t}(E)ρ_{s}) for t > s ≥ 0 and a Borel set E ∈ C_{s,t} (note that the probability does not depend on C_{s}, and depends linearly on ρ_{s}). It follows that 𝔼 (ρ_{t}|C_{s}, ρ_{s}) = ∫_{C_{s,t}} p_{s,t}(ρ_{s}, C)μ_{s,t}(ρ_{s}, dC) = E_{s,t}(C_{s,t})ρ_{s} = T_{t−s}(ρ_{s}) by 3.4(f), so,

(4.7)  𝔼 (ρ_{t}|C_{s}, ρ_{s}) = T_{t−s}(ρ_{s})

for rational s, t such that t ≥ s ≥ 0. The other part of 3.4(f) gives

(4.8)  ℙ (C_{t} ∩ (s, t) = ∅|C_{s}, ρ_{s}) = tr (x_{t−s}ρ_{s}x_{t−s}^{∗}) .

The space V has its dual space V^{∗}, identified with the space of all Hermitian operators A : H → H; the natural bilinear form is of course A, ρ ↦ tr (Aρ). The semigroup (T_{t}) on V has its dual semigroup T_{t}^{∗} on V^{∗}; we have tr (ρT_{t}^{∗}(A)) = tr (T_{t}(ρ)A) = 𝔼 tr (X(t)ρX^{∗}(t)A) = 𝔼 tr (ρX^{∗}(t)AX(t)), thus,

(4.9)  T_{t}^{∗}(A) = 𝔼 (X^{∗}(t)AX(t))

for A ∈ V^{∗} and t ≥ 0. Note that V is also a subset of V^{∗}. The following property is peculiar for quantum dynamical semigroups generated by unitary Brownian motions:

(4.10)  If A ∈ V  then T_{t}^{∗}(A) ∈ V and tr (T_{t}^{∗}(A)) = tr (A) ,

which follows from (4.9), since tr (X^{∗}(t)AX(t)) = tr ((X(t))^{−1}AX(t)) = tr (A), and A ≥ 0 implies X^{∗}(t)AX(t) ≥ 0.

4.11 Lemma. Let Q_{n} ∈ V^{∗} be finite-dimensional projections, and ε_{n} → 0 positive numbers. Then the set K = {ρ ∈ V_{1}^{+} : tr (Q_{n}ρ) ≥ 1 − ε_{n} for n = 1, 2, …} is compact.

Proof. For every n the set {Q_{n}ρQ_{n} : ρ ∈ V_{1}^{+}} is finite-dimensional and bounded. It suffices to prove that ∥ρ−Q_{n}ρQ_{n}∥ ≤ 2\sqrt{ε_{n}} for all ρ ∈ K. Each ρ ∈ V_{1}^{+} may be represented as ρ = 𝔼 ρ_{h} for some random vector h ∈ H, ∥h∥ = 1. We have ∥ρ_{h} − Q_{n}ρ_{h}Q_{n}∥^{2} = ∥ρ_{h}−ρ_{Q_{n}h}∥^{2} ≤ 4∥h−Q_{n}h∥^{2} = 4(1−∥Q_{n}h∥^{2}) = 4(1−tr (Q_{n}ρ_{h})), therefore ∥ρ−Q_{n}ρQ_{n}∥^{2} ≤ (𝔼 ∥ρ_{h} − Q_{n}ρ_{h}Q_{n}∥)^{2} ≤ 𝔼 ∥ρ_{h} − Q_{n}ρ_{h}Q_{n}∥^{2} ≤ 4(1 − 𝔼 tr (Q_{n}ρ_{h})) = 4(1 − tr (Q_{n}ρ)) ≤ 4ε_{n} for ρ ∈ K.

4.12 Lemma. Let S_{n} ⊂ V^{∗} be compact sets of compact operators, and ε_{n} → 0 positive numbers. Then the set K = {ρ ∈ V_{1}^{+} : max_{A∈S_{n}} tr (Aρ) ≥ 1 − ε_{n} for n = 1, 2, …} is compact.

Proof. Compactness of an operator A ∈ S_{n} implies existence of a finite-dimensional projection Q_{n}^{(A)} such that A ≤ Q_{n}^{(A)} + ε_{n} · 1_{H}. Compactness of the set S_{n} allows to choose a single finite-dimensional projection Q_{n} such that A ≤ Q_{n} + ε_{n} · 1_{H} for all A ∈ S_{n}. Then tr (Aρ) ≤ tr (Q_{n}ρ)+ε_{n} for all A ∈ S_{n}, therefore tr (Q_{n}ρ) ≥ max_{A∈S_{n}} tr (Aρ)−ε_{n} ≥ 1−2ε_{n} for all ρ ∈ K. Lemma 4.11 completes the proof.

4.13 Lemma. For every ρ_{0} ∈ V_{1}^{+}, t ∈ (0, ∞), and ε > 0 there is a compact set K ⊂ V_{1}^{+} such that ℙ (ρ_{s} ∈ K for all rational s ∈ [0, t]) ≥ 1 − ε, where ρ_{s} is defined by (4.6).

Proof. Let Q ∈ V^{∗} be a finite-dimensional projection; introduce a martingale M(s) = 𝔼 (tr (Qρ_{t})|C_{s}, ρ_{s}). Using (4.7), M(s) = tr (QT_{t−s}(ρ_{s})) = tr (Q_{s}ρ_{s}) where Q_{s} = T_{t−s}^{∗}(Q). Note that Q_{s} is continuous in s (the same argument as in the proof of 3.4(e)), therefore {Q_{s} : s ∈ [0, t]} is a compact set. Taking into account that M(s) ≤ 1 always, we have ℙ (inf_{s∈[0,t]} M(s) < 1 − ε) ≤ \frac{1}{c}(1 − M(0)) = \frac{1}{c}(1 − tr (QT_{t}(ρ_{0}))).

Choose finite-dimensional projections Q_{1}, Q_{2}, … such that tr (Q_{n}T_{t}(ρ_{0})) → 1. Introduce compact sets S_{n} = {T_{t−s}^{∗}(Q_{n}) : s ∈ [0, t]} ⊂ V^{∗}; each T_{t−s}^{∗}(Q_{n}) is a compact operator due to (4.9). Choose c_{n} → 0, c_{n} > 0 such that δ_{n} → 0, where δ_{n} = (1 − tr (Q_{n}T_{t}(ρ_{0})))/c_{n}. Martingales M_{n}(s) = tr (T_{t−s}^{∗}(Q_{n})ρ_{s}) satisfy ℙ (inf_{s∈[0,t]} M_{n}(s) < 1 − c_{n}) ≤ δ_{n}. Choosing a subsequence we can get ∑ δ_{n} < ∞; then ℙ (E_{n}) ≥ 1 − ε_{n}, where E_{n} = {ω : M_{k}(s) ≥ 1 − c_{k} for all s ∈ [0, t] and k = n, n + 1, …} and ε_{n} = δ_{n} + δ_{n+1} + … → 0. Within E_{n} we have tr (T_{t−s}^{∗}(Q_{k})ρ_{s}) ≥ 1 − c_{k}, therefore max_{A∈S_{k}} tr (Aρ_{s}) ≥ 1 − c_{k} for k = n, n + 1, … and all rational s ∈ [0, t]. Lemma 4.12 ensures that the set K_{n} = {ρ ∈ V_{1}^{+} : max_{A∈S_{k}} tr (Aρ) ≥ 1 − c_{k} for k = n, n + 1, …} is compact. So, ℙ (ρ_{s} ∈ K_{n} for all rational s ∈ [0, t]) ≥ ℙ (E_{n}) ≥ 1 − ε_{n} → 1.

For a given ρ_{0} ∈ V_{1}^{+} construct compact subsets K_{1} ⊂ K_{2} ⊂ … of V_{1}^{+} such that ℙ (ρ_{t} ∈ K_{n} for all rational t ∈ [0, n]) ≥ 1 − 2^{−n} for all n. Introduce first exit times τ_{n} = inf{t : ρ_{t} ∉ K_{n}}; these are stopping (Markov) times. (They may be irrational, which is harmless, since we do not need ρ_{τ_{n}}.) We have

(4.14)  τ_{1} ≤ τ_{2} ≤ …; τ_{n} → ∞; ρ_{t} ∈ K_{n} for t < τ_{n}; K_{1}, K_{2}, … are compact.

Denote f_{r}(ρ) = tr (x_{r}ρx_{r}^{∗}) = tr (x_{r}^{∗}x_{r}ρ) for ρ ∈ V_{1}^{+}, r ∈ [0, ∞), then ℙ (C_{s+r} = C_{s}|C_{s}, ρ_{s}) = f_{r}(ρ_{s}). We have f_{r}(ρ) → 1 for r → 0, since f_{r}(ρ_{h}) = tr (ρ_{x_{r}h}) = ∥x_{r}h∥^{2} → ∥h∥^{2} (here ρ_{h} is the one-dimensional operator, ρ_{h}x = (x, h)h). Functions f_{r}(·) are in fact linear functionals on V of norm ≤ 1, therefore the convergence f_{r}(·) → 1 must be uniform on compact subsets of V_{1}^{+}. We may choose rational r_{n} > 0 such that f_{r_{n}}(ρ) ≥ 1/2 for ρ ∈ K_{n}; so,

(4.15)  ℙ (C_{t+r_{n}} = C_{t}|C_{t}, ρ_{t}) ≥ \frac{1}{2} for n = 1, 2, … and all rational t ∈ [0, τ_{n}) .

Consider the number card (C_{t+r_{n}} ∩ (t, τ_{n})) (maybe, +∞) of points in C_{t+r_{n}} belonging to the (maybe empty) interval (t, τ_{n}).

4.16 Lemma. For every k, n = 1, 2, … and every rational t ∈ [0, ∞)

ℙ (card (C_{t+r_{n}} ∩ (t, τ_{n})) ≥ k) ≤ 2^{−k} .

Proof. Fix n and t; take an integer m, divide the interval (t, t + r_{n}) into m intervals I_{1}, …, I_{m} of equal length, and consider the (random) number N_{m} of intervals I_{i} such that I_{i} ∩ C_{t+r_{n}} ∩ (t, τ_{n}) ≠ ∅. It suffices to prove that ℙ (N_{m} ≥ k) ≤ 2^{−k} for all m, since N_{m} → card (C_{t+r_{n}} ∩ (t, τ_{n})) for m → ∞, and moreover, the convergence is monotone for m = 1, 2, 4, 8, … Further, it suffices to prove that ℙ (N_{m} ≥ k + 1|N_{m} ≥ k) ≤ 1/2 for k = 0, 1, 2, … However, this fact follows from (4.15) applied for t, t + (1/m)r_{n}, t + (2/m)r_{n}, …, t + ((m − 1)/m)r_{n} by the standard argument with a Markov time, which is legal, since the Markov time takes on only a finite number of (rational) values.

So, C_{t} ∩ [0, τ_{n}] has a finite intersection with every interval of length r_{n}, therefore C_{t} ∩ [0, τ_{n}] is finite; however, τ_{n} → ∞, thus C_{t} is finite for all t almost sure, that is,

(4.17)  μ_{0,t}(ρ_{0}, C_{finite}) = 1 .

We have tr (E_{0,t}(C_{finite})ρ_{0}) = 1; tr (E_{0,t}(C \ C_{finite})ρ_{0}) = 0; by positivity (3.4b2), E_{0,t}(C \ C_{finite})ρ_{0} = 0. However, ρ_{0} is arbitrary; so,

(4.18)  E_{0,t}(C \ C_{finite}) = 0 .

Combining it with (3.6) we get μ_{f}(C \ C_{finite}) = 0 for each f of the form f = (X(t)h_{2}, h_{1}). It means that all such f belong to L̂_{finite}. By Corollary 2.13, the noise is linearizable. Thus, the main part of Theorem 1.6 is achieved: the infinite-dimensional unitary group is Brown subordinate to the Hilbert space. The converse holds by the two following facts.

4.19 Note. Let G_{1}, G_{2} be Polish groups. If there exists a continuous one-one homomorphism G_{1} → G_{2}, then G_{1} is Brownian subordinate to G_{2}. (The proof is immediate.)

4.20 Note. There exists a continuous one-one homomorphism from (the additive group of) the Hilbert space to the unitary group.

Proof. Let ξ_{1}, ξ_{2}, … be i.i.d. N(0, 1) random variables. Each c = (c_{1}, c_{2}, …) ∈ l_{2} determines a random variable exp(i ∑ c_{k}ξ_{k}), and the corresponding multiplication operator U(c) on the space of all square integrable random variables. The operator U(c) is unitary, and the map c ↦ U(c) is a continuous one-one homomorphism.

5. The commutative case

Proof of Theorem 1.8. Let G be a commutative Polish group, and X a Brownian motion in G. Due to Corollary 2.13 it suffices to prove that φ(X(t)) ∈ L̂_{finite} for every bounded Borel function φ : G → ℝ and t ∈ [0, ∞). That is, we have to prove that μ_{f}(C_{finite}) = 1, where f = φ(X(t)), ∥f∥_{L_{2}} = 1.

We have f = φ(X_{0,t/2} + X_{t/2,t}) = φ(X_{t/2,t} + X_{0,t/2}) due to commutatitity of G. It follows that the joint probability distribution (under μ_{f}) of C ∩ (0, \frac{t}{2}) and C ∩ (\frac{t}{2}, t) is the same as for C ∩ (\frac{t}{2}, t) and C ∩ (0, \frac{t}{2}). That is, μ_{f} is invariant under the piecewise linear transformation α : (0, t) → (0, t), α(s) = s + \frac{t}{2} for s ∈ (0, \frac{t}{2}), α(s) = s − \frac{t}{2} for s ∈ (\frac{t}{2}, t). Likewise, μ_{f} is invariant under the group of all invertible piecewise linear transformations of (0, t) having derivative = 1 on each piece (the group acts naturally on C_{0,t} modulo negligible sets).

Take an integer n, divide the interval (0, t) into n subintervals of length t/n, and consider the probability distribution (under μ_{f}) of the number k of subintervals that intersect C; denote the probabilities by a_{0}, a_{1}, …, a_{n} (a_{0} + … + a_{n} = 1). On the other hand, for any ε ∈ (0, 1) consider δ(ε) = μ_{f}{C : C ∩ [0, εt] ≠ ∅}; we have δ(ε) → 0 for ε → 0 due to 2.3(a). Given the number k (of subintervals that intersect C), all the \binom{n}{k} possibilities are equiprobable due to the invariance property of μ_{f}, and the conditional probability of \{C : C ∩ [0, \frac{m}{n}t] = ∅\} is \binom{n−m}{k}/\binom{n}{k}. So,

δ(\frac{m}{n}) = 1 − ∑_{k=0}^{n} a_{k} \begin{matrix}( & {n−m \atop k} & ) \\ ( & {n \atop k} & ) \end{matrix}

for m = 1, …, n. However, \binom{n−m}{k}/\binom{n}{k} = \frac{n−m}{n} · \frac{n−m−1}{n−1} ·. . .· \frac{n−m−k+1}{n−k+1} ≤ (\frac{n−m}{n})^{k}, therefore

δ(\frac{m}{n}) ≥ 1 − ∑ a_{k}(\frac{n−m}{n})^{k} = ∑ a_{k}(1 − (\frac{n−m}{n})^{k}), and

a_{k} + … + a_{n} ≤ \frac{δ(\frac{m}{n})}{1 − (\frac{n−m}{n})^{k}}

for all k and m; the left-hand side does not depend on m.

Take an ε ∈ (0, 1) and consider n = 2, 4, 8, … together with m_{n} such that m_{n}/n < ε, m_{n}/n → ε (while k does not depend on n). Note that a_{k} + … + a_{n} (which should be denoted rigorously by a_{k}^{(n)} + … + a_{n}^{(n)}) tends to μ_{f}{C : card (C) ≥ k}. We get

μ_{f}\{C : card (C) ≥ k\} ≤ \frac{δ(ε)}{1 − (1 − ε)^{k}}

for all k = 1, 2, … and ε ∈ (0, 1); the left-hand side does not depend on ε. It remains to choose ε_{k} → 0 such that (1 − ε_{k})^{k} ≤ 1/2, getting μ_{f}{C : card (C) ≥ k} ≤ 2δ(ε_{k}) → 0, and so, μ_{f}(C_{finite}) = 1.

Proof of Corollary 1.9. A separable F-space is Brown subordinate to the Hilbert space due to Theorem 1.8; it suffices to prove that the Hilbert space is Brown subordinate to every infinite-dimensional separable F-space. It follows immediately from Note 4.19 and the following fact.

5.1 Note. Let Δ be the Hilbert space, and F an infinite-dimensional F-space. Then there exists a continuous one-one linear operator Δ → F.

Proof. Take x_{1}, x_{2}, … ∈ F spanning an infinite-dimensional subspace. Define T : l_{2} → F by T (ξ_{1}, ξ_{2}, …) = ∑ ξ_{n}c_{n}x_{n}, where c_{n} tend to 0 fast enough, then T is continuous. Take Δ = l_{2} ⊖ T^{−1}(0) and restrict T to Δ.

The following fact will not be used formally, but is worth to be mentioned. In particular, it shows that the proof of Corollary 1.10 is simpler than it seems; path-to-path correspondences used are in fact point-to-point.

5.2 Lemma. Let G be a commutative Polish group, X a Brownian motion in G, Δ the Hilbert space, and (X, Y) a Brownian motion in G × Δ such that F_{t}^{X} ⊂ F_{t}^{Y} for all t ∈ [0, ∞). Then for every t ∈ [0, ∞), X(t) is measurable w.r.t. the σ-field generated by Y (t).

Proof. It suffices to prove that the random variable f = φ(X(t)) is measurable w.r.t. the σ-field generated by Y (t) for every bounded Borel function φ : G → ℝ. We may assume that Δ is a space of sequences, and Y (t) = (B_{1}(t), B_{2}(t), …), where B_{1}, B_{2}, … are independent standard Brownian motions (in ℝ). Like every element of L_{2}(F_{t}^{Y}), f can be decomposed into multiple Itô integrals,

f = ∑_{n=0}^{∞} ∫ …∫ ∑_{k_{1},...,k_{n}} f̂_{k_{1},...,k_{n}}(t_{1}, …, t_{n}) dB_{k_{1}}(t_{1}) … dB_{k_{n}}(t_{n}) .

Remind the invariance argument used in the proof of Theorem 1.8: piecewise linear transformations of (0, t) (with derivative = 1 on each piece) act on paths of (X, Y) by measure preserving transformations, leaving invariant X(t) and f. Therefore, f̂_{k_{1},...,k_{n}}(t_{1}, …, t_{n}) remains unchanged, when the transformation acts on each of t_{1}, …, t_{n}. It means that f̂_{k_{1},...,k_{n}}(t_{1}, …, t_{n}) does not depend on t_{1}, …, t_{n}, it is a constant! Thus, the stochastic integral is a polynomial of Y (t) only.

Denote by γ the standard Gaussian measure on the space ℝ^{∞} of all sequences of reals; that is, γ is the joint distribution of a sequence of i.i.d. N(0, 1) random variables. Given a Polish group G, we introduce the set L_{0}(γ, G) of all equivalence classes of γ-measurable functions X : ℝ^{∞} → G, the equivalence being the equality γ-almost everywhere.

5.3 Lemma. Let G be a commutative Polish group. For any X ∈ L_{0}(γ, G) the following three properties are equivalent.

(a) There exists Y ∈ L_{0}(γ, G) such that X(u) + X(v) = Y (2^{−1/2}(u + v)) for γ ⊗ γalmost all pairs (u, v) ∈ ℝ^{∞} × ℝ^{∞}.

(b) For each a ∈ l_{2} there exists g ∈ G such that X(u + a) = X(u) + g for γ-almost all u ∈ ℝ^{∞}.

(c) There exist a subgroup E ⊂ ℝ^{∞} of γ-full measure, a homomorphism X_{1} : E → G, and an element g ∈ G such that X(u) = X_{1}(u) + g for γ-almost all u.

Proof. (c) ⟹(a): X(u) + X(v) = X_{1}(u) + g + X_{1}(v) + g = X_{1}(u + v) + 2g; we let Y (w) = X_{1}(2^{1/2}w) + 2g, taking into account that 2^{1/2}w ∈ E for γ-almost all w.

(a) ⟹(b): X(u + a) + X(v − a) = Y (2^{−1/2}(u + v)) = X(u) + X(v), therefore X(u + a) − X(u) = X(v) − X(v − a) for γ ⊗ γ-almost all (u, v), which means that both functions are constant γ-almost everywhere.

(b) ⟹(c). The function X_{0} : l_{2} → G defined by X(u + a) = X(u) + X_{0}(a) is a homomorphism. If a_{1}, a_{2}, … ∈ l_{2}, a_{n} → 0 weakly in l_{2}, then X_{0}(a_{n}) → 0_{G} for n → ∞, since X(· + a_{n}) → X(·) in probability. Introduce projections P_{n} : ℝ^{∞} → ℝ^{n} ⊂ l_{2} ⊂ ℝ^{∞}, P_{n}(ξ_{1}, ξ_{2}, …) = (ξ_{1}, …, ξ_{n}) = (ξ_{1}, …, ξ_{n}, 0, 0, …), and functions X_{n} = X − X_{0} ∘ P_{n}, that is, X_{n}(ξ_{1}, ξ_{2}, …) = X(ξ_{1}, ξ_{2}, …) − X_{0}(ξ_{1}, …, ξ_{n}, 0, 0, …), then X_{n}(u + a) = X_{n}(u) for all a ∈ ℝ^{n}, which means that X_{n}(ξ_{1}, ξ_{2}, …) depends only on ξ_{n+1}, ξ_{n+2}, …

Functions of the form φ_{n} ∘ P_{n} (for all n and all measurable φ_{n} : ℝ^{n} → G) are dense in L_{0}(γ, G) (equipped with the convergence in probability). Choose φ_{n} such that φ_{n}∘P_{n} → X in probability. We have φ_{n} ∘ P_{n} − X → 0 in probability. Choose ε_{n} → 0 such that γ{u : ρ(0, φ_{n}(P_{n}u) − X(u)) ≤ ε_{n}} ≥ 1 − ε_{n}. The probability of the event ρ(0, φ_{n}(P_{n}u) − X(u)) ≤ ε_{n} is the expectation of the conditional probability of the same event, given the first n coordinates ξ_{1}, …, ξ_{n} of u = (ξ_{1}, ξ_{2}, …). Given n, we may choose ξ_{1}, …, ξ_{n} such that the conditional probability at (ξ_{1}, …, ξ_{n}) is ≥ 1 − ε_{n}. However, φ_{n}(P_{n}u) − X(u) = φ_{n}(P_{n}u) − X_{0}(P_{n}u) − X_{n}(u). Denote g_{n} = φ_{n}(ξ_{1}, …, ξ_{n}) − X_{0}(ξ_{1}, …, ξ_{n}, 0, 0, …), then γ{u : ρ(0, g_{n} − X_{n}(u)) ≤ ε_{n}} ≥ 1 − ε_{n}; the conditioning is omitted, since X_{n}(ξ_{1}, ξ_{2}, …) depends only on ξ_{n+1}, ξ_{n+2}, … So, there exist g_{1}, g_{2}, … ∈ G such that X_{n} − g_{n} → 0 in probability, which means that X − X_{0} ∘ P_{n} − g_{n} → 0, that is, X_{0} ∘ P_{n} + g_{n} → X in probability, when n → ∞. The following trick shows that g_{n} → g for some g.

Consider three measure preserving maps α, α_{1}, α_{2} : (ℝ^{∞} × ℝ^{∞}, γ ⊗ γ) → (ℝ^{∞}, γ) defined by α_{1}(u, v) = \frac{1}{2}u+ \frac{\sqrt{3}}{2}v, α_{2}(u, v) = \frac{1}{2}u− \frac{\sqrt{3}}{2}v, α(u, v) = u, then α_{1}(u, v)+α_{2}(u, v) = α(u, v). For any a, b ∈ l_{2} we have (X ∘α_{1}+X ∘α_{2}−X ∘α)(u+a, v+b)−(X ∘α_{1}+X ∘α_{2}−X ∘ α)(u, v) = X_{0}(α_{1}(a, b))+X_{0}(α_{2}(a, b))−X_{0}(α(a, b)) = X_{0}(α_{1}(a, b)+α_{2}(a, b)−α(a, b)) = 0, which means that the function X∘α_{1}+X∘α_{2}−X∘α is constant; X(α_{1}(u, v))+X(α_{2}(u, v))− X(α(u, v)) = g for γ ⊗ γ-almost all (u, v). However, X ∘ α_{1} = lim_{n}(X_{0} ∘ P_{n} ∘ α_{1} + g_{n}), and the same for α_{2}, α. We have g = lim_{n}(X_{0}∘P_{n}∘α_{1}+g_{n}+X_{0}∘P_{n}∘α_{2}+g_{n}−X_{0}∘P_{n}∘α−g_{n}) = lim_{n}(X_{0} ∘ P_{n} ∘ (α_{1} + α_{2} − α) + g_{n}) = lim_{n} g_{n}.

So, X_{0} ∘ P_{n} + g → X in probability, when n → ∞. We choose n_{k} → ∞ such that X_{0} ∘ P_{n_{k}} + g → X almost sure. The set of all u such that lim_{k} X_{0}(P_{n_{k}}(u)) exists, is a subgroup E of ℝ^{∞}, γ(E) = 1, and the limit is the needed homomorphism X_{1} : E → G.

The convergence X_{0} ∘ P_{n} + g → X, obtained in the proof above, may be compared with other results [18, 31, 9]. There, convergence almost sure is established for linear spaces; here — convergence in probability, for commutative groups. Also, Lemma 5.3 may be compared with the study of “quasi-additive functionals” in [9]. There, maps G → ℝ are considered; here — maps ℝ^{∞} → G.

5.4 Corollary. Let G be (the additive group of) a separable F-space. Then, in Condition (c) of Lemma 5.3, the subgroup E can be chosen to be a linear subspace, and the homomorphism X_{1} — a linear map.

Proof. The function X_{0} : l_{2} → G is linear, since it is a continuous homomorphism between F-spaces. Functions X_{0} ∘ P_{n_{k}} are linear, therefore their limit X_{1} is linear, and its domain E is a linear subspace.

All reasonable definitions of a Gaussian measure on a separable Banach space are evidently equivalent, which cannot be said about separable F-spaces and Polish groups (see [9]).

A symmetric Gaussian measure in the sense of Fernique is a probability measure μ on a separable F-space F such that the product measure μ ⊗ μ on F ⊕ F is invariant under the following group of transformations: T_{φ}(x, y) = ( x cos φ − y sin φ, x sin φ + y cos φ ) for x, y ∈ F, φ ∈ ℝ.

A Gaussian measure in the sense of Bernstein is a probability measure μ on a commutative Polish group G such that the product measure μ ⊗ μ on G × G turns into some product measure ν_{1}⊗ν_{2} on G×G under the following transformation: (x, y) ↦ (x−y, x+y) for x, y ∈ G.

If μ is a symmetric Gaussian measure in the sense of Fernique, then μ (as well as any shift of μ) is also Gaussian in the sense of Bernstein.

A Gaussian (convolution) semigroup is a family (μ_{t})_{t∈[0,∞)} of probability measures μ_{t} on a Polish group G, produced by some Brownian motion X(·) in G, in the sense that μ_{t} is the distribution of X(t) for each t. (See [8].)

Let us define a constructively Gaussian measure as a probability measure μ on a commutative Polish group G, that can be represented as the distribution of some X ∈ L_{0}(γ, G) satisfying (equivalent) conditions (a–c) of Lemma 5.3.

A constructively Gaussian measure is a Gaussian measure in the sense of Bernstein. (Indeed, 5.3(c) transfers Bernstein property of γ into Bernstein property of μ.)

A constructively Gaussian measure on a separable F-space is a shift of a symmetric Gaussian measure in the sense of Fernique. (Indeed, 5.3(c) and 5.4 transfer Fernique property of γ into Fernique property of μ shifted by (−g).)

If a measure on a separable F-space is contained in a Gaussian semigroup, then it is constructively Gaussian (which is shown below by means of Theorem 1.8). The following fact is thus obtained.

5.5 Corollary. If a measure μ on a separable F-space is contained in a Gaussian semigroup, then some shift of μ is a symmetric Gaussian measure in the sense of Fernique, and μ is a Gaussian measure in the sense of Bernstein.

A Brownian motion in a separable F-space F determines a measure μ on the space C_{0}([0, ∞), F ) of all continuous functions x : [0, ∞) → F such that x(0) = 0. The space C_{0}([0, ∞), F ), equipped with the topology uniform on finite intervals, is also a separable F-space.

Corollary 1.10 (final formulation). For every Brownian motion in a separable Fspace F, the corresponding measure μ on the space C_{0}([0, ∞), F ) is constructively Gaussian.

The corresponding Gaussian semigroup (μ_{t}) results from μ by applying evaluation maps x(·) ↦ x(t). Each evaluation map is a continuous linear map C_{0}([0, ∞), F ) → F, therefore it sends a constructively Gaussian measure into another constructively Gaussian measure. Thus, 5.5 follows from 1.10.

Proof of Corollary 1.10. Let X be a Brownian motion in a separable F-space F. By Theorem 1.8, there exists a Brownian motion (X, Y) in F × Δ, where Δ is the Hilbert space, such that F_{t}^{X} = F_{t}^{Y} for all t. Take a Borel function f : C_{0}([0, ∞), Δ) → C_{0}([0, ∞), F ) such that f(Y) = X almost sure (here X is treated as a random variable Ω → C_{0}([0, ∞), F )).

Introduce two independent copies (X_{1}, Y_{1}) and (X_{2}, Y_{2}) of the Brownian motion (X, Y). It is easy to see that the process (X_{1} + X_{2}, Y_{1} + Y_{2}) is also a Brownian motion. Moreover, the process t ↦ (X_{1}(t/2) + X_{2}(t/2), Y_{1}(t/2) + Y_{2}(t/2)) is another copy of (X, Y), since (X_{1}(t/2), X_{2}(t/2)) is distributed like (X(t/2), X(t) − X(t/2)), and the same for (X, Y) pairs. For notational convenience, introduce h : [0, ∞) → [0, ∞) by h(t) = t/2; we see that (X_{1} ∘ h + X_{2} ∘ h, Y_{1} ∘ h + Y_{2} ∘ h) is distributed like (X, Y). Therefore, f(Y_{1} ∘h+Y_{2} ∘h) = X_{1} ∘h+X_{2} ∘h almost sure. However, f(Y_{1}) = X_{1} and f(Y_{2}) = X_{2}, thus, f(Y_{1} ∘ h + Y_{2} ∘ h) = f(Y_{1}) ∘ h + f(Y_{2}) ∘ h. Define g : C_{0}([0, ∞), Δ) → C_{0}([0, ∞), F ) by g(y) = f(2^{1/2}y ∘ h) ∘ h^{−1}, then g(2^{−1/2}(Y_{1} + Y_{2})) = f(Y_{1}) + f(Y_{2}) almost sure. It means that f satisfies Condition (a) of Lemma 5.3. Though, the corresponding measure γ is not the standard Gaussian measure on ℝ^{∞}, rather it is some Gaussian measure on C_{0}([0, ∞), Δ). However, it is well-known that each Gaussian measure on a locally convex F-space is linearly isomorphic mod 0 to the standard Gaussian measure on ℝ^{∞}. Due to Lemma 5.3, f satisfies its Condition (c), therefore the distribution of f(Y) = X is constructively Gaussian.

5.6 Note. Linearity of F was not used (multiplications by 2^{±1/2} were made in Δ, not in F). Thus, Corollary 1.10 (and its proof) remains true for all commutative Polish groups.

Appendix

Proof of Lemma 3.1. First, let H be a Hilbert space of finite or countable dimension, X a Brownian motion in the unitary group U(H), and (T_{t}) the semigroup on V defined by (3.3). We extend (T_{t}) to the complexification of V, the space of all (not only Hermitian) trace-class operators, by complex linearity; still, (3.3) holds. Introduce a notation for one-dimensional operators on H: for any h_{1}, h_{2} ∈ H

⟨⟨h_{1}, h_{2}⟩⟩ : H → H , ⟨⟨h_{1}, h_{2}⟩⟩h = (h, h_{2})h_{1} for h ∈ H .

(Physicists denote it by |h_{1}⟩⟨h_{2}|.) Note some general rules:

tr (⟨⟨h_{1}, h_{2}⟩⟩) = (h_{1}, h_{2}) ,
(⟨⟨h_{1}, h_{2}⟩⟩)^{∗} = ⟨⟨h_{2}, h_{1}⟩⟩ ,
A⟨⟨h_{1}, h_{2}⟩⟩ = ⟨⟨Ah_{1}, h_{2}⟩⟩ , ⟨⟨h_{1}, h_{2}⟩⟩A = ⟨⟨h_{1}, A^{∗}h_{2}⟩⟩ for A : H → H ,
⟨⟨h_{1}, h_{2}⟩⟩⟨⟨h_{3}, h_{4}⟩⟩ = (h_{3}, h_{2})⟨⟨h_{1}, h_{4}⟩⟩ ,
tr (A⟨⟨h_{1}, h_{2}⟩⟩) = (Ah_{1}, h_{2}) .

Using the Hilbert-Schmidt scalar product (B, C)_{HS} = tr (BC^{∗}) for trace class operators B, C : H → H, calculate a matrix element for T_{t}: (T_{t}⟨⟨h_{1}, h_{2}⟩⟩, ⟨⟨h_{3}, h_{4}⟩⟩)_{HS} = 𝔼 tr (X(t)⟨⟨h_{1}, h_{2}⟩⟩X^{∗}(t)⟨⟨h_{4}, h_{3}⟩⟩) = 𝔼 tr (⟨⟨X(t)h_{1}, X(t)h_{2}⟩⟩⟨⟨h_{4}, h_{3}⟩⟩) = 𝔼 (h_{4}, X(t)h_{2})(X(t)h_{1}, h_{3}) = 𝔼 (X(t), ⟨⟨h_{3}, h_{1}⟩⟩)_{HS}(⟨⟨h_{4}, h_{2}⟩⟩, X(t))_{HS}. In particular,

(A.1)  (T_{t}⟨⟨h_{1}, h_{2}⟩⟩, ⟨⟨h_{1}, h_{2}⟩⟩)_{HS} = 𝔼 (X(t), ρ_{1})_{HS}(ρ_{2}, X(t))_{HS} ,

where ρ_{k} = ⟨⟨h_{k}, h_{k}⟩⟩.

From now on, H is assumed to be finite-dimensional. We have diffusion processes A(t) in Δ and X(t) = exp(iA(t)) in U(H); here Δ is the linear space of all Hermitian operators on H. We may write A(t) = A_{0}(t) + λ(t) · 1_{H}, tr A_{0}(t) = 0, λ(t) ∈ ℝ. For small t,

\begin{matrix}X(t) = 1 + iA(t) − \frac{1}{2}A^{2}(t) + o(t) ; \\ (X(t), ρ)_{HS} = 1 + i(A(t), ρ)_{HS} − \frac{1}{2}(A^{2}(t), ρ)_{HS} + o(t) \end{matrix}

for ρ = ⟨⟨h, h⟩⟩, ∥h∥ = 1. Therefore, for ρ_{k} = ⟨⟨h_{k}, h_{k}⟩⟩, ∥h_{k}∥ = 1 (k = 1, 2),

𝔼 (X(t), ρ_{1})_{HS}(ρ_{2}, X(t))_{HS} =
= 1 + i(𝔼 A(t), ρ_{1})_{HS} − i(ρ_{2}, 𝔼 A(t))_{HS}+
+ 𝔼 (A(t), ρ_{1})_{HS}(ρ_{2}, A(t))_{HS} − \frac{1}{2}(𝔼 A^{2}(t), ρ_{1})_{HS} − \frac{1}{2}(ρ_{2}, 𝔼 A^{2}(t))_{HS} + o(t) .

Separating real and imaginary terms (you see, A(t) and ρ_{k} are Hermitian) and using (A.1) we conclude that the following two expressions are uniquely determined by the semigroup (T_{t}):

\frac{d}{dt}|_{t=0}(𝔼 A(t), ρ_{1} − ρ_{2})_{HS} ; (A.2)

\frac{d}{dt}|_{t=0}𝔼 (A(t), ρ_{1})_{HS}(A(t), ρ_{2})_{HS} − \frac{1}{2} \frac{d}{dt}|_{t=0}(𝔼 A^{2}(t), ρ_{1} + ρ_{2})_{HS} . (A.3)

Operators of the form ρ_{1} − ρ_{2} belong to the space Δ_{0} of all traceless Hermitian operators, and Δ_{0} is spanned by such operators. Thus, (A.2) means that the following is uniquely determined by (T_{t}):

(A.4)  \frac{d}{dt}|_{t=0}𝔼 A_{0}(t) .

Consider four expressions of the form (A.3): first, exactly as (A.3), that is, for the pair (ρ_{1}, ρ_{2}); second, the same for another pair, (ρ_{3}, ρ_{4}); also, for (ρ_{1}, ρ_{4}) and (ρ_{3}, ρ_{2}), the last two with the minus sign. Summing the four, we get

(A.5)  𝔼 (A(t), ρ_{1} − ρ_{3})_{HS}(A(t), ρ_{2} − ρ_{4})_{HS} .

It means that the whole bilinear form describing the spread (at t = 0+) of A_{0}(t) in Δ_{0} is determined by (T_{t}). Together with (A.4) it proves that the generator of the Brownian motion (det X(t))^{−1/d}X(t) = exp(iA_{0}(t)) in SU(d) is uniquely determined by (T_{t}), which completes the proof of 3.1(a).

Turn to the semigroup (x_{t}), x_{t} = 𝔼 X(t). We have x_{t} = 1 + i𝔼 A(t) − \frac{1}{2}𝔼 A^{2}(t) + o(t); separating real and imaginary terms we conclude that the following two expressions are uniquely determined by (x_{t}):

\frac{d}{dt}|_{t=0}𝔼 A(t) , (A.6)

\frac{d}{dt}|_{t=0}𝔼 A^{2}(t) . (A.7)

However, A(t) = A_{0}(t) + λ(t) · 1_{H}, and λ(t) = (1/d)tr A(t). Taking the trace of (A.6), we find \frac{d}{dt} |_{t=0}𝔼 λ(t). Further, 𝔼 A^{2}(t) = 𝔼 A_{0}^{2}(t) + 2𝔼 λ(t)A_{0}(t) + 𝔼 λ^{2}(t) · 1_{H}, and 3.1(a) ensures that 𝔼 A_{0}^{2}(t) is determined by (T_{t}). Thus, the following is uniquely determined by (x_{t}) and (T_{t}):

(A.8)  2\frac{d}{dt}|_{t=0}𝔼 λ(t)A_{0}(t) + 1_{H} · \frac{d}{dt}|_{t=0}𝔼 λ^{2}(t) .

The two terms can be separated by taking the trace. So, (x_{t}) and (T_{t}) determine

(A.9)  \frac{d}{dt}|_{t=0}𝔼 λ(t) , \frac{d}{dt}|_{t=0}𝔼 λ^{2}(t) , \frac{d}{dt}|_{t=0}𝔼 λ(t)A_{0}(t) ;

these are all the infinitesimal characteristics of A (or X), besides characteristics of A_{0} given by 3.1(a).

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