Within and Beyond the Reach

of Brownian Innovation

Boris Tsirelson

Abstract. Given a system whose time evolution is random, we often try to describe it as a deterministic system under independent random influences. Doing so, we reduce complicated statistical correlations to a complicated but deterministic mechanism, and a stochastic but uncorrelated noise. That is the idea of innovation. The corresponding mathematics is surprisingly interesting.

1991 Mathematics Subject Classification: 60G07; 60H10, 60J65. Keywords and Phrases: innovation, filtration, cosiness, noise.

1. The name of the game

An innovation is a real-time transformation of a noise into a given random process.

Out of the four terms, only one, “random process”, is standard. The notion of a real-time transformation was introduced repeatedly, and used under various names: “lifting” (of a filtered probability space) [19, (7.1–7.3)], “Hypothèse (H)” [7, Sect. 2.4], “extension” (of a filtered probability space) [22, Chap. 2, Def. 7.1], [8, Def. 6.1], with no name [44, 17.3.1(a)], [2, Lemma 7(c)], “morphism” (from one filtration to another) [33, Def. 1.1], “immersion” (of one filtration into another) [4], “orthogonal factor” (of a reverse filtration) [15, Sect. 2]. My favorite “real-time transformation” appeared in [33].

A noise in the discrete-time framework amounts to an independent sequence (of random variables or σ-fields), or a product (of a sequence of probability spaces). For continuous time, the classical white noise is a special case of a noise as defined in [34, Def. 1.1]; see also “factored probability spaces” [13], “measure factorizations” [36, Def. 1.2], and “product measures” (on a factorized Borel space) [36, Def. 2.4].

Innovation processes are well-known in filtering theory (see [5, Sect. 8]). A farreaching generalization is the “innovation” introduced here. In the discrete-time framework, innovation appeared as “standard extension” (of a reverse filtration) [8, p. 885], “generating parametrization” [28, Sect. 2], [26, Def. 2.1], “substandard representation” [15, Sect. 2]. My favorite “innovation” appeared in [26].

2. Trivial cases

Let μ be a probability measure on a space X. (Usually X = ℝ or ℝ^{n}, but it may be a finite set, a complete separable metric space, a standard Borel space.) Every such μ can be represented as the image of the Lebesgue measure U(0, 1) under a measurable map f : (0, 1) → X. Let U be a random variable distributed uniformly on (0, 1) (in symbols U ∼ U(0, 1)), then f(U) ∼ μ. Of course, μ does not determine f uniquely; f(g(U)) ∼ μ for all measure preserving g : (0, 1) → (0, 1). So, every X -valued random variable Y is distributed like some X = f(U).

Consider a discrete time random process Y = (Y_{t})_{t∈T}, assuming for now that T is finite, T = {1, …, n}; thus Y is just n random variables Y_{1}, …, Y_{n}, and its distribution is a measure μ on X^{n}. Let U_{1}, …, U_{n} be independent U(0, 1) random variables. Choose f_{1} : (0, 1) → X such that f_{1}(U_{1}) is distributed like Y_{1}. For each y_{1} ∈ X consider the conditional distribution of Y_{2} given that Y_{1} = y_{1} (I omit trivial reservations) and choose f_{2}(·, y_{1}) accordingly. Introduce X_{1} = f_{1}(U_{1}), X_{2} = f_{2}(U_{2}, X_{1}), then the pair (X_{1}, X_{2}) is distributed like (Y_{1}, Y_{2}). Continuing the process, we get functions f_{1}, …, f_{n} and random variables X_{1}, …, X_{n} such that

(2.1) X_{1} = f_{1}(U_{1}), X_{2} = f_{2}(U_{2}, X_{1}), …, X_{n} = f_{n}(U_{n}, X_{n−1}, …, X_{1}) , (X_{1}, …, X_{n}) is distributed like (Y_{1}, …, Y_{n}) .

That is the innovation: at a time t ∈ T the process X takes on a value X_{t} produced by a deterministic mechanism f_{t} out of two sources: the past (X_{1}, …, X_{t−1}) of the process, and the current value U_{t} of a noise. Note that each U_{t} is used only once (formulas like X_{2} = f_{2}(U_{2}, U_{1}, X_{1}) are disallowed), and U_{1}, …, U_{n} are independent. The uniform distribution of U_{t} is only conventional; in Sect. 4 we prefer the normal distribution. Note also the large choice available on each stage when constructing f_{1}, …, f_{n}.

Example. Let (Y_{t})_{t∈T} be a process with independent increments, having assumed that X = ℝ or another group. We may choose an innovation of the form

(2.2)  X_{t} = g_{t}(U_{t}) + X_{t−1} .

The simple form (2.2) seems to be decidedly preferable to (2.1) for such processes, which is a delusion, to be refuted in Sect. 3.

The distribution of X = (X_{1}, …, X_{n}) is the given μ. Consider, however, the joint distribution of X and U. We have

(2.3)  𝔼 ( φ(X_{1}, …, X_{n}) | U_{1}, …, U_{t} ) = 𝔼 ( φ(X_{1}, … X_{n}) | X_{1}, … X_{t} )

for all t = 1, …, n and all bounded Borel functions φ : X^{n} → ℝ. Forecasting the future of the process X, we want to know the past of X only, and not the past of U. In other words, (X_{t+1}, …, X_{n}) and (U_{1}, …, U_{t}) are conditionally independent, given (X_{1}, …, X_{t}). *

Consider the σ-field F_{X}(t) generated by X_{1}, …, X_{t}; clearly, F_{X}(t) ⊂ F_{U}(t) for all t, that is, F_{X} ≤ F_{U}, where F_{X} = (F_{X}(t))_{t∈T} is the filtration generated by X. Writing (2.3) in the form 𝔼 ( ξ | F_{U}(t) ) = 𝔼 ( ξ | F_{X}(t) ) for F_{X}(n)-measurable ξ, note that 𝔼 ( ξ | F_{X}(t) ) is the general form of an F_{X}-martingale; so,

(2.4)  M(F_{X}) ⊂ M(F_{U}) ,

* Though, (2.1) stipulates more: (X_{t+1}, U_{t+1}, …, X_{n}, U_{n}) and (U_{1}, …, U_{t}) are conditionally independent, given (X_{1}, …, X_{t}). where M(F) is the set of all F-martingales. Relation (2.4) implies F_{X} ≤ F_{U}, and is much stronger; try X_{2} = f_{2}(U_{2}, U_{1}, X_{1}) instead of f_{2}(U_{2}, X_{1}) and you’ll find (2.4) violated but F_{X} ≤ F_{U} is still valid.

The following definition is formulated in terms of processes, but only their distributions are relevant. Still, T = {1, …, n}.

2.6 Definition. A real-time transformation of a random process V = (V_{1}, …, V_{n}) into another process W = (W_{1}, …, W_{n}) is a two-component process (V′, W′) = ((V′_{1}, W′_{1}), …, (V′_{n}, W′_{n})) such that V′ is distributed like V, W′ is like W, and for each t = 1, …, n, W′_{t} is equal to a function of V′_{1}, …, V′_{t}, and two vectors (V′_{1}, …, V′_{t}) and (W′_{t+1}, …, W′_{n}) are conditionally independent given (W′_{1}, …, W′_{t}).

Reformulations via (2.3), (2.4) and generalizations for infinite T are left to the reader. Nothing new emerges for an infinite increasing sequence of time moments, t ∈ T = ℕ = {1, 2, 3, …}. Still, an innovation is constructed step-by-step: f_{1}, then f_{2}, and so on ad infinitum. The same holds for every countable ordinal number, that is, every countable linearly ordered set T that contains no infinite strictly decreasing sequences.

3. Decreasing sequences are highly non-trivial

The following two examples show an astonishing phenomenon: some information appears magically, from thin air; see [25, p. 156], [43, p. 136], [10] and references therein.

The first example: X_{t} = ±1 for t ∈ ℤ are i.i.d. equiprobable random signs, U_{t} = X_{t}/X_{t−1}; then U_{t} are i.i.d. equiprobable random signs, also. Thus, X is both a process with independent values, and a process with independent increments in the multiplicative group {−1, +1}. The equality X_{t} = U_{t}X_{t−1} should be an innovation of the process X by the noise U. However, it is not; X contains more information than U, since U determines X only up to an overall sign. The missing information should be a kind of initial value, X_{−∞}; however, any function of the germ (tail) of X at −∞ is either constant almost sure, or nonmeasurable, which is the well-known tail triviality.

The second example is the “eternal” (stationary) Brownian motion in a circle (or any other compact Lie group). Let (B(t))_{t∈[0,∞)} be the standard Brownian motion in ℝ, and α a random variable, uniform on (0, 1) and independent of (B(t))_{t∈[0,∞)}. Consider the complex-valued process X(t) = exp (2πiα + iB(t)). The process (X(t))_{t∈[0,∞)} is stationary. Therefore, it has a unique (in distribution) extension (X(t))_{t∈ℝ}, the eternal motion. Multiplicative increments U_{t} = X_{t}/X_{t−1} for t ∈ ℤ should innovate the process (X(t))_{t∈ℤ}. However, they do not, for the same reason as in the first example: they stay invariant under transformations of the form (X(t))_{t∈ℝ} ↦ (e^{iφ}X(t))_{t∈ℝ}.

About notation: ergodic people, being more light-hearted toward the time arrow than probabilists, prefer (X′_{1}, X′_{2}, …), where X′_{1} = X_{−1}, X′_{2} = X_{−2}, …, to (…, X_{−2}, X_{−1}). Accordingly, dependence on the past turns into dependence on larger indices t [8], [16], [28], [26], [15]. I adhere to the probabilistic school, [44], [4], [9], [10], choosing T = (−ℕ) = {…, −2, −1}.

Every process Y = (Y_{t})_{t∈T} is distributed like some process X satisfying X_{t} = f_{t}(U_{t}; X_{t−1}, X_{t−2}, …) for some Borel functions f_{t} and independent U_{t}. It follows that M(F_{X}) ⊂ M(F_{X,U}), but we need M(F_{X}) ⊂ M(F_{U}). The two-component process (U, X) is a real-time transformation of U into X if and only if F_{X} ≤ F_{U}. Chaining f_{t}, f_{t−1}, …, f_{s+1} we get f_{s,t} such that X_{t} = f_{s,t}(U_{t}, …, U_{s+1}; X_{s}, X_{s−1}, …). However, we need f_{−∞,t} such that X_{t} = f_{−∞,t}(U_{t}, U_{t−1}, …). That is possible if and only if the influence of X_{s}, X_{s−1}, … on f_{s,t}(U_{t}, …, U_{s+1}; X_{s}, X_{s−1}, …) disappears in the limit s → −∞. Tail triviality is necessary but not sufficient. Both examples shown above are tail trivial, and satisfy X_{t} = U_{t} … U_{s+1}X_{s}. Given U, the influence of X_{s} on X_{t} is strong, irrespective of s. Thus, the equality X_{t} = U_{t}X_{t−1} fails to give an innovation.

Despite the strong influence of X_{s} on X_{t}, these X_{s}, X_{t} are (statistically) independent in the first example, and asymptotically independent (for s → −∞) in the second example. The strong dependence characterizes the specific way of using U_{t} (namely, X_{t} = U_{t}X_{t−1}), that is, the parametrization (f_{t})_{t∈T} rather than the process X itself. Is there a better parametrization for the same process? For the first example, the answer is evidently positive. Here, the conditional distribution of X_{t}, given the past, does not depend on the past. The parametrization X_{t} = U_{t}X_{t−1} is bad because it introduces an unnecessary dependence on the past. A good parametrization is simply X_{t} = U_{t}, which surely is an innovation. For the second example, restricted to t ∈ ℤ, the conditional distribution of X_{t}, given the past, depends on X_{t−1}. However, such distributions (corresponding to different values of X_{t−1}) overlap. A good parametrization uses the overlap for reducing dependence on the past. In continuous time, an innovation for the eternal motion is constructed [10] by inventing a coupling for processes differing in remote past. They are forced to coalesce, which never happens under the bad parametrization X_{t} = U_{t}X_{t−1} of the form (2.2). That is the refutation of the delusion mentioned after (2.2).

Is there an innovation for an arbitrary tail-trivial process (X_{t})_{t∈(−ℕ)} ? The answer is negative, which fact is “highly non-trivial and remarkable” [26], “deep and surprising” [15]. The first example, admitting no innovation, was discovered in the context of ergodic theory [37]. There are more examples of ergodic flavor [38], [29], [39], [28], [21], and of probabilistic flavor [8], [17], [14], [26], [4], [9]. The example of [8], furthered in [17], [14], [26], [4], is strikingly close to the sequence of i.i.d. equiprobable random signs; namely, the product measure is replaced with an equivalent (that is, mutually absolutely continuous) measure.

Some criteria for existence of an innovation, outlined in [37], [39], are elaborated in [15]. There, “substandardness” is our “existence of innovation”, while “product type” is stronger, stipulating that U_{t} is a function of X_{t}, X_{t−1}, … In such a case one says that U_{t} is exactly the new information furnished by X at t (though it depends on the chosen innovation). “Substandardness” implies “product type” provided that the conditional distribution of X_{t} given the past, is nonatomic [15].

4. Cosiness

Cosiness is a useful necessary condition for existence of an innovation. (Is it also sufficient? I do not know.) Cosiness emerged in [33, Def. 2.4] for continuous time and in [4, Sect. 4] for discrete time, the latter with a reservation that “there is a whole range of possible variations” of the definition; one of the variations follows. Still, T = (−ℕ) = {…, −2, −1}, and processes are X -valued.

4.1 Definition. A random process (X_{t})_{t∈T} is cosy, if for each ε > 0 and each bounded Borel function φ : X^{T} → ℝ there exists a two-component random process (Y, Z) = ((Y_{t}, Z_{t}))_{t∈T} such that

(a) ((Y, Z), Y ) and ((Y, Z), Z) are real-time transformations of (Y, Z) into X;

(b) 𝔼|φ(Y) − φ(Z)| < ε;

(c) there exists δ ∈ (0, 1) such that for all bounded Borel functions ψ, χ : X^{T} → ℝ,

(𝔼|ψ(Y)χ(Z)|)^{2−δ} ≤ (𝔼|ψ(Y)|^{2−δ})(𝔼|χ(Z)|^{2−δ}) .

Some comments. Condition (a) implies that each of the two processes Y, Z is distributed like X; thus, (Y, Z) is a joining of two copies of X, possessing the “real time” property M(Y) ⊂ M(Y, Z), M(Z) ⊂ M(Y, Z) (recall (2.4)). Condition (b) means that Y, Z are close, since φ may be one-one. Condition (c) means that Y, Z are “independent a little”, since it is always satisfied for δ = 0 and equivalent to independence of Y, Z for δ = 1.

4.2 Theorem. [4, Lemma 6 and Corollary 3] A non-cosy process admits no innovation.

The idea of a proof. Assume that X has an innovation; X is distributed like Y, Y_{t} = f_{−∞,t}(U_{t}, U_{t−1}, …), U = (U_{t})_{t∈T} being a sequence of independent N (0, 1) random variables. (This time we prefer the normal distribution N (0, 1) to U(0, 1).) Take another sequence V = (V_{t})_{t∈T} of independent N (0, 1) random variables such that U, V are independent. Introduce W_{t} = U_{t} cos ε + V_{t} sin ε, and let Z_{t} = f_{−∞,t}(W_{t}, W_{t−1}, …).* Condition (c) follows from the celebrated hypercontractivity theorem (pioneered by Nelson, see [24, Sect. 3])!

The first example of a non-cosy process in discrete time is given in [4, Th. 1]; it appears that the method of [8] produces non-cosy processes. It is interesting to know, whether “ergodic” examples [37], [38], [29], [39], [28], [21] are also noncosy, or not. Another non-cosy discrete-time filtration [9] is the restriction of a continuous-time filtration to a discrete set on the time axis.

5. Applications to continuous time

An X -valued process (X_{t})_{t∈T}, T = (−ℕ) = {…, −2, −1}, generates its filtration F_{X} = (F_{X}(t))_{t∈T}. The family (F_{X}(2t))_{t∈T} is also a filtration; it is generated by the X^{2}-valued process (Y_{t})_{t∈T}, Y_{t} = (X_{2t−1}, X_{2t}). If X admits an innovation, then the amalgamated process Y also does. The same applies for any infinite subset T_{1} ⊂ T. If X is tail-trivial and T_{1} is sparse enough, then Y admits an innovation, see [15, Th. 1.18] and references therein.

A continuous process (X_{t})_{t∈[0,∞)} generates its filtration F_{X} = (F_{X}(t))_{t∈[0,∞)}. Choosing a sequence (t_{k})_{k∈(−ℕ)}, t_{k} ∈ [0, ∞), t_{k−1} < t_{k}, inf t_{k} = 0, we get a

* Which is anticipated in [23]. discrete-time filtration (F_{X}(t_{k}))_{k∈(−ℕ)}, generated by the amalgamated process (Y_{k})_{k∈(−ℕ)}, Y_{k} = (X_{t})_{t∈[t_{k−1},t_{k}]}. If Y admits no innovation, then X also admits no innovation, for any reasonable definition of continuous-time innovations. Some continuous-time problems are solved in that way.

Documenta Mathematica · Extra Volume ICM 1998 · III · 311–320

The effect of “information from thin air” (see Sect. 3) can be reproduced by the stochastic differential equation

(5.1)  dX_{t} = dB_{t} + v ( t, (X_{s})_{s∈[0,t]} ) dt

with a bounded drift v, if v is chosen properly. Then (5.1) fails to innovate X, which means that the equation has no strong solution. That is the “celebrated and mysterious” [25, V.3.18, p. 155] example, constructed in [32] and investigated in [5], [30], [43], [23], [10]. The eternal Brownian motion in a circle, mentioned in Sect. 3, can be obtained from X by a real-time transformation and a deterministic time change that maps [0, ∞) onto ℝ [10]. The same process X is a strong solution of the stochastic differential equation

(5.2)  dX_{t} = σ ( t, (X_{s})_{s∈[0,t]} ) dB_{t} + v ( t, (X_{s})_{s∈[0,t]} ) dt

for some σ(…) = ±1 [10] (see also [16]). Once again, a clever parametrization is better than the straightforward parametrization.

One of the processes admitting no innovation, mentioned in Sect. 3, leads to a more ingenious drift v in (5.1); the corresponding (continuous) process X has no innovation, which means that it cannot be the strong solution of any equation of the form (5.2) [8]; see also [17], [14], [26], [4]. The drift is not bounded, but I believe that it can be made bounded. “Dreadfully complicated, their construction is almost as incredible as the existence result itself” [4]. Is it really a complicated construction? In fact, the drift is not constructed “by hands”, it is chosen at random. It is a random drift; here “random” is interpreted like the second “random” in the phrase “random walk in a random environment”. Thus, it is a typical drift in the same sense as a nowhere differentiable Brownian sample path is a typical function. Few parameters are adjusted by authors, such as order of magnitude, and depth of dependence on the past, both depending on time in a simple prescribed way.

There exists a pure martingale admitting no innovation [9].

6. From stochastic analysis to stochastic topology

Some continuous-time phenomena have no (evident) discrete-time counterpart. For example, Brownian motion cannot be transformed in real time into a Poisson process. A non-Gaussian stable process cannot be transformed into Brownian motion. The m-dimensional Brownian motion can be transformed into the ndimensional Brownian motion if and only if m ≥ n, which may be treated as the starting point of stochastic topology, the theory of filtration invariants of random processes.* A diffusion process with smooth nondegenerate coefficients in an ndimensional smooth manifold is equivalent to the n-dimensional Brownian motion

* A useful classification claimed in [27, Th. 7] appeared to be not exhaustive [8, Sect. 6]. in the sense that their filtrations are isomorphic; in other words, the two processes can be connected by an invertible real-time transformation. What happens in presence of singularities of the topology or the coefficients? Few results are available; they are based on stochastic analysis (Itô formula, local times, …). All negative results are based on continuous-time cosiness [33, Def. 2.4]. Brownian motion of finite or countable dimension is cosy [33, Lemma 2.5]. A cosy process cannot be transformed in real time to a non-cosy process [33, Lemma 2.6]. Therefore, all non-cosy processes are beyond the reach of Brownian innovation.

Two well-known diffusion processes in ℝ are singular at the origin (x = 0, not t = 0 as in Sect. 5). The skew Brownian motion (see [20]) has a singular drift at 0, and is equivalent to the usual Brownian motion [20]. The sticky Brownian motion (see [41]) is slowed down at 0; its filtration is non-cosy [42].

Consider n rays (say, on the plane) with a single common point, the origin. There is a natural diffusion process Z_{n} on the union of the rays; Z_{2} is the usual Brownian motion, Z_{1} is the reflecting Brownian motion; Z_{3}, Z_{4}, … are so-called Walsh’s Brownian motions [40], [3]. Such processes arise when considering small random perturbations of Hamiltonian dynamical systems [18] and some other topics [40], [3]. Processes Z_{1} and Z_{2} are equivalent (Lévy, Skorokhod). Nevertheless, Walsh’s Brownian motions are non-cosy [33, Th. 4.13] (see also [11], [2]), which solves Problem 2 of [3].

Interestingly, stochastic topology can be of help to the classical (nonstochastic) analysis. Consider three non-intersecting domains in ℝ^{n}. If they are smoothly bounded, then points of trilateral contact are evidently rare among boundary points. It was conjectured for irregular domains, that the infimum of the three corresponding harmonic measures must vanish [6, Sect. 6], [12, Problem a]. In terms of the Martin boundary: its natural projection to the topological boundary is at most 2 to 1 almost everywhere. However, the best result of classical analysis is “at most 10 to 1” [6]. The final result “2 to 1” is achieved via stochastic topology [33, Th. 7.4]. A challenge for classical analysis!

So, some characteristic of ℝ^{n} (or any smooth manifold) as a harmonic space, is equal to 2 irrespective of dimension, but exceeds 2 in presence of branching points. The nameless characteristic has its counterpart in stochastic topology, named splitting multiplicity. Introduced in [3, Def. 4.2], it was hibernating till the birth of cosiness. Every cosy process is of splitting multiplicity 2 (or 1, if it is degenerate) [2], while Walsh’s Brownian motion Z_{n}, n > 2, is of splitting multiplicity n [2]. Splitting multiplicity is invariant under measure changes and time changes [2], while cosiness is not [4], [9].

7. White noise versus black noises

In discrete time we have no choice of noises for innovation; a noise is a sequence of independent random variables, each having a non-atomic distribution. In continuous time, the classical theory of processes with independent increments tells us that in general, a noise consists of a Gaussian component (a finite or countable collection of independent white noises) and a Poissonian component. The latter is useless for innovating diffusion processes. The former can innovate only cosy processes. Thus, Walsh’s Brownian motion is beyond the reach of classical innovation.

We may turn to Brownian motions (defined as continuous processes with stationary independent increments) on more general groups. In that aspect, finitedimensional Lie groups are equivalent to ℝ^{n}. The Polish group of all unitary operators on the (separable) Hilbert space, equipped with the strong operator topology, is equivalent to (the additive group of) the Hilbert space [34, Th. 1.6]. (Interestingly, the proof involves continuous tensor products and continuous quantum measurements.) A commutative Polish group cannot give more [34, Th. 1.8].

The system of coalescing independent one-dimensional Brownian motions [1], [31, Sect. 2], is a limiting case of a coalescing stochastic flow. The system generates a two-parametric family of σ-fields (F_{s,t})_{s<t} that shares with the white noise the following property:

(7.1)  F_{r,s} ⊗ F_{s,t} = F_{r,t} whenever r < s < t;

that is, F_{r,s} and F_{s,t} are independent and, taken together, generate F_{r,t}. Nevertheless, (F_{s,t})_{s<t} supports no white noise (nor a Poisson process); it means that there is no Brownian motion (B_{t})_{t∈[0,∞)} such that B_{t} − B_{s} is F_{s,t}-measurable for all intervals (s, t) ⊂ [0, ∞) [35]. Thus, (F_{s,t})_{s<t} is a black noise as defined in [34, Sect. 1]. It is predictable [34, Def. 1.12] in the sense that its filtration (F_{0,t})_{t∈[0,∞)} supports only continuous martingales. In fact, the filtration is Brownian! Therefore, that black noise still cannot innovate Walsh’s Brownian motion.

One more example of a black noise is available [36, Sect. 5]. Does it generate a cosy filtration? I do not know.

7.2 Problem. Can a predictable noise (see [34, Defs. 1.1, 1.12]) generate a noncosy filtration?

If the answer is positive, another problem follows.

7.3 Problem. Can Walsh’s Brownian motion be innovated by some predictable noise?

7.4 Problem. Can a noise generate a cosy but non-Brownian filtration?*

References

[1] R. Arratia, Coalescing Brownian motions, and the voter model on Z, manuscript, Univ. of Southern California, 1985.

[2] M.T. Barlow, M. Émery, F.B. Knight, S. Song, M. Yor, Autour d’un théorème de Tsirelson sur des filtrations browniennes et non browniennes, Lect. Notes Math. (Séminaire de Probabilités XXXII), Springer, Berlin, 1686 (1998), 264–305.

[3] M. Barlow, J. Pitman, M. Yor, On Walsh’s Brownian motions, Lect. Notes Math. (Séminaire de Probabilités XXIII), Springer, Berlin, 1372 (1989), 275–293. (MR 91a:60204)

* It was claimed in the abstract of my talk, that it never happens to Brownian motions in Polish groups. However, I withdraw the claim, since my proof was wrong. Sorry.

Innovation 319

[4] S. Beghdadi-Sakrani, M. Émery, On certain probabilities equivalent to coin-tossing, d’après Schachermayer, Lect. Notes Math. (Séminaire de Probabilités XXXIII), Springer, Berlin, to appear.

[5] V.E. Beneš, Nonexistence of strong nonanticipating solutions to stochastic DEs: implications for functional DEs, filtering, and control, Stoch. Proc. Appl. 5:3 (1977), 243–263. (MR 56#16788)

[6] C.J. Bishop, A characterization of Poissonian domains, Arkiv för Matematik 29:1 (1991), 1–24. (MR 93a:31011)

[7] P. Brémaud, M. Yor, Changes of filtrations and of probability measures, Z. Wahrsch. Verw. Gebiete 45:4 (1978), 269–295. (MR 80h:60062)

[8] L. Dubins, J. Feldman, M. Smorodinsky, B. Tsirelson, Decreasing sequences of σ-fields and a measure change for Brownian motion, Ann. Probab. 24:2 (1996), 882–904. (MR 97g:60106)

[9] M. Émery, W. Schachermayer, private communication, March 1998.

[10] M. Émery, W. Schachermayer, private communication, April 1998.

[11] M. Émery, M. Yor, Sur un théorème de Tsirelson relatif à des mouvements browniens corrélés et à la nullité de certains temps locaux, Lect. Notes Math. (Séminaire de Probabilités XXXII), Springer, Berlin, 1686 (1998), 306–312.

[12] A. Eremenko, B. Fuglede, M. Sodin, Harmonic measure for three disjoint domains in ℝ^{n}, in “Linear and Complex Analysis Problem Book 3” (V.P. Havin, N.K. Nikolski, eds.) (1994), 323–325.

[13] J. Feldman, Decomposable processes and continuous products of probability spaces, J. Funct. Anal. 8 (1971), 1–51. (MR 44#7617)

[14] J. Feldman, ε-close measures producing nonisomorphic filtrations, Ann. Probab. 24:2 (1996), 912–915. (MR 97g:60108)

[15] J. Feldman, Decreasing sequences of measurable partitions: product type, standard, and prestandard, Ergodic Theory and Dynamical Systems (to appear).

[16] J. Feldman, M. Smorodinsky, Simple examples of non-generating Girsanov processes, Lect. Notes Math. (Séminaire de Probabilités XXXI), Springer, Berlin, 1655 (1997), 247–251.

[17] J. Feldman, B. Tsirelson, Decreasing sequences of σ-fields and a measure change for Brownian motion.II, Ann. Probab. 24:2 (1996), 905–911. (MR 97g:60107)

[18] M. Freidlin, Markov Processes and Differential Equations: Asymptotic Problems, Birkhäuser Verlag, Basel, 1996. (MR 97f:60150)

[19] R.K. Getoor, M.J. Sharpe, Conformal martingales, Invent. Math. 16 (1972), 271– 308. (MR 46#4603)

[20] J.M. Harrison, L.A. Shepp, On skew Brownian motion, Ann. Probab. 9:2 (1981), 309–313. (MR 82j:60144)

[21] D. Heicklen, C. Hoffman, T, T^{−1} is not standard, Ergodic Theory and Dynamical Systems (to appear).

[22] N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, second edition, North-Holland, 1989. (MR 90m:60069)

[23] J.F. Le Gall, M. Yor, Sur l’equation stochastique de Tsirelson, Lect. Notes Math. (Séminaire de Probabilités XVII), Springer, Berlin, 986 (1983), 81–88. (MR 86j:60132)

[24] E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973), 211–227. (MR 49#8556)

[25] L.C.G. Rogers, D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2: Itô Calculus, Wiley, New York, 1987. (MR 89k:60117)

320 B. Tsirelson

[26] W. Schachermayer, On certain probabilities equivalent to Wiener measure, d’après Dubins, Feldman, Smorodinsky and Tsirelson, Lect. Notes Math. (Séminaire de Probabilités XXXIII), Springer, Berlin, to appear.

[27] A.V. Skorokhod, Stochastic processes in infinite-dimensional spaces, Proc. Internat. Cong. Math. (A.M. Gleason, ed.) 163–171. Amer. Math. Soc., Providence, RI, 1987. (In Russian.) (MR 89e:60081)

[28] M. Smorodinsky, Processes with no standard extension, Israel J. Math. (to appear).

[29] A.M. Stepin, On entropy invariants of decreasing sequences of measurable partitions, Funct. Anal. Appl. 5:3 (1971), 237–240 (transl. from Russian).

[30] D.W. Stroock, M. Yor, On extremal solutions of martingale problems, Ann. Sci. École Norm. Sup. (4) 13:1 (1980), 95–164. (MR 82b:60051)

[31] B. Tóth, W. Werner, The true self-repelling motion, Probab. Theory Related Fields (to appear).

[32] B.S. Tsirel’son, An example of a stochastic differential equation having no strong solution, Theory Probab. Appl. 20:2 (1975), 416–418 (transl. from Russian). (MR 51#11654)

[33] B. Tsirelson, Triple points: from non-Brownian filtrations to harmonic measures, Geom. Funct. Anal. (GAFA) 7 (1997), 1096–1142.

[34] B. Tsirelson, Unitary Brownian motions are linearizable, MSRI Preprint No. 1998-027, math.PR/9806112.

[35] B. Tsirelson, Brownian coalescence as a black noise, manuscript in preparation.

[36] B.S. Tsirelson, A.M. Vershik, Examples of nonlinear continuous tensor products of measure spaces and non-Fock factorizations, Reviews in Mathematical Physics 10:1 (1998), 81–145.

[37] A.M. Veršik, Decreasing sequences of measurable partitions and their applications, Soviet Math. Dokl. 11:4 (1970), 1007–1011 (transl. from Russian). (MR 42#3258)

[38] A.M. Vershik, Continuum of pairwise nonisomorphic diadic sequences, Funct. Anal. Appl. 5 (1971), 182–184 (transl. from Russian). (MR 43#7593).

[39] A.M. Vershik, Theory of decreasing sequences of measurable partitions, St. Petersburg Math. J. 6:4 (1995), 705–761 (transl. from Russian). (MR 96b:28018)

[40] J.B. Walsh, A diffusion with a discontinuous local time, Astérisque, 52–53 (1978), 37–45.

[41] J. Warren, Branching processes, the Ray-Knight theorem, and sticky Brownian motion, Lect. Notes Math. (Séminaire de Probabilités XXXI), Springer, Berlin, 1655 (1997), 1–15.

[42] J. Warren, On the joining of sticky Brownian motion, technical report of the dept. of statistics, university of Warwick, 1998.

[43] M. Yor, Tsirel’son’s equation in discrete time, Probab. Theory Related Fields 91:2 (1992), 135–152. (MR 93d:60104)

[44] M. Yor, Some Aspects of Brownian Motion, part II: Some Recent Martingale Problems, Birkhäuser Verlag, Basel, 1997. (MR 98e:60140)

Boris Tsirelson

School of Mathematics Tel Aviv University Tel Aviv 69978, Israel email: tsirel@math.tau.ac.il