Around Tsirelson's equation, or: The evolution process may not explain everything

We present a synthesis of a number of developments which have been made around the celebrated Tsirelson's equation (1975), conveniently modified in the framework of a Markov chain taking values in a compact group $ G $, and indexed by negative time. To illustrate, we discuss in detail the case of the one-dimensional torus $ G=\bT $.

1 Introduction 1 • ). The contents of this paper, which were presented at the Meeting "Dynamical Systems and Randomness" at IHP, Paris (May 15th, 2009), were motivated mainly by the authors' desire ( [22], [1] and [13]) to understand deeply Tsirelson's equation [3]. This equation shall be discussed in Section 4, while, as a preparation for the main part of the paper, we shall discuss the stochastic equation: on a compact group G, where k varies in −N, (ξ k ) k≤0 is the "evolution process", and (η k ) k≤0 is the unknown process, both taking values in G. We believe that equation (1) is the "right" abstraction of Tsirelson's equation, as shown in Section 4.
More precisely, for every k ≤ 0, the law of ξ k is a given probability µ k on G, and the ξ k 's are assumed independent. We shall denote the sequence (µ k ) k≤0 simply by µ. It is immediate that (η k ) k≤0 is a Markov chain, the transitions of which are given by The problem is that, as k varies in −N, there is no initial state ("at time −∞"), and the study of P µ , the set of the laws of all the solutions of (1), necessitates some care.
We shall also be interested in ex(P µ ), the set of all extremal points of the compact set P µ , as well as in S µ , the set of laws P of "strong" solutions, i.e.: under P , F η k ⊂ F ξ k , hence, since from (1), there is the identity: then: P ∈ S µ iff: under P , F η k = F ξ k , for every k. A number of natural questions now arise: given µ = (µ k ) k≤0 , a) is there existence for (1)?, i.e.: P µ = ∅. b) is there uniqueness?, i.e., ♯(P µ ) = 1. c) is there a strong solution?, i.e.: S µ = ∅.
We shall see that these different questions may be answered very precisely, in particular if G = T ≃ [0, 1) is the one-dimensional torus, in terms of criteria on µ.

• ). Interpreting Tsirelson's equation:
• Before proceeding, we would like to give a "light" interpretation of equation (1), we mean one not to be taken too seriously!: η k describes the "state of the universe" at time k; this state is "created" by the state at time k − 1, followed by the action of the "evolution" ξ k . The main question is: can today's state of the universe, i.e.: η 0 , be explained solely from the evolution process?, an almost metaphysical question... We shall see that the answer(s), in terms of µ, are somewhat paradoxical... • Then our "interpretation" also allows us to justify our quite general choice of the probabilities (µ k ) k≤0 . Indeed, today's "historians of the universe" see the evolution process at work, say, for times j ∈ [K, 0], for some large negative K (K decreases as "today" increases...). But, they have no knowledge beyond that K, hence, we need to make the most general assumptions on the (µ k )'s to understand all possible cases... Thus, depending on the choice of µ, mathematicians give an answer as to how much the evolution process determines the present state, but this choice remains to be made! Here, G = T ≃ [0, 1); η k = exp(2iπθ k ); ξ k = exp(2iπg k ), with θ k ∈ [0, 1), and g k Gaussian, centered, E[g 2 k ] = σ 2 k . The answers to our previous questions are radically different depending on whether The case: k≤0 σ 2 k = ∞.
• For any fixed k, η k is uniformly distributed on the torus, (or θ k is uniform on [0, 1)), independent from the evolution sequence (ξ j ) j≤0 .

−→
k→−∞ V with V independent of the evolution (ξ j ), Thus, here in the framework of (4), there is nonuniqueness iff there is a strong solution, which is indeed a puzzling result.
We give the main arguments of proof when: We first show that: ∀k, η k is uniformly distributed, i.e.: for p ∈ Z, p = 0, we obtain: Thus, θ k is uniform on [0, 1), i.e.: η k is uniform on the torus. This reinforces, as we can show, likewise: Then, letting N → ∞: Hence, the independence of η k , for any fixed k, from the evolution process (ξ j ) j≤0 . In consequence, the law of (η k ) k≤0 is uniquely determined by this independence property.
The triviality of F η −∞ will be explained (in the next section, Theorem 3.2) by a general result, i.e.: the triviality of F η −∞ under any P ∈ ex(P µ ), but, here, there is only one solution!!

The general group framework -Questions and facts -
Let G be a general compact group; there is the uniform distribution (= Haar measure), and we now take up the discussion of the general questions a), b), c) stated in the Introduction.
Proof. It follows from Kolmogorov's extension theorem, since for any k, we may consider the law U . . , η 0 ]; then, for k < l and p k,l , the obvious projection, we find that: l . So, the laws (U Thus, there is always existence; and, we may ask the 2 questions: b) is there uniqueness?, i.e.: ♯(P µ ) = 1?; c) does a strong solution exist?, i.e.: S µ = ∅?
Let us make the following table: uniqueness strong solution holds fails Discussion: We immediately rule out C 0 , since, from Theorem 3.1, under C 0 , the unique solution P * µ is not strong. Thus, there remains to discuss the trichotomy: C 1 -C 2 -C 3 We now state several general results: The diagonal operator: τ g : (η 0 k ) k → (η 0 k g) k acts transitively over ex(P µ ), i.e.: τ g : P 0 ∈ ex(P µ ) → τ g (P 0 ) ∈ ex(P µ ) and the mapping is surjective.
iii). Any solution (η k ) k≤0 may be represented as: with V G-valued and independent of (η 0 k ) k . This yields directly the Krein-Milman integral representation: for any given extremal solution (η 0 k ) k .
To get a good feeling /introduction/ for the following discussion, we recall a result of Csiszár ([5]): i.e., the "almost" convergence in law of infinite products of independent random variables. ). Let (ξ j ) j≤0 be our evolution sequence. There exists a sequence (α l , l → −∞) of deterministic elements of G such that, for any fixed k ∈ −N, the sequence (ξ k ξ k−1 · · · ξ l α l ) l<k (16) converges in law, as l → −∞.
We give two illustrations of Theorem 3.3.
1 • ). As a first illustration of Theorem 3.3, let us go back to the Gaussian set-up of Section 2, where we now consider more generally with g k Gaussian, with variance σ 2 k , and mean m k . Then, we may choose as "centering sequence".
2 • ). To illustrate further Theorem 3.3, or may be, more accurately, point 1) of Theorem 3.4 below, we may consider the case where all the laws µ k are the same; then, Stromberg [18] (see also Collins [4]) showed that, for ν a given probability on G, ν * n converges to Haar measure as soon as the smallest subgroup which contains the support of ν is equal to G.
We may now present a characterization of C 1 and C 2 .
2). There exists a strong solution iff there exists a sequence (α l ) of deterministic elements of G such that the products ξ k ξ k−1 · · · ξ l α l converge a.s. as l → −∞.
Then, every extremal solution is strong and is the law of the a.s. limit of (ξ k ξ k−1 · · · ξ l α l g), for some g ∈ G.
Again, to illustrate Theorem 3.4, we may consider the general Gaussian hypothesis made after Theorem 3.3; clearly, uniqueness holds iff k σ 2 k = ∞, whereas C 2 holds iff k σ 2 k < ∞. Note that C 3 never occurs in this set-up. To give a full discussion of the trichotomy, we go back to G = T ≃ [0, 1). We introduce: Then, there is the Proposition 3.1 (Yor [22]). Z µ is a subgroup of Z; hence, there exists a unique integer p µ ≥ 0 such that Z µ = p µ Z.
This Proposition now allows us to discuss fully the trichotomy C 1 -C 2 -C 3 .

Remark.
A special case of a) above is when ε j = j for some random variable γ with absolutely continuous density. This is called "Poincaré roulette wheel, leading to equidistribution"; see [8,Theorem 3.2] for detail.

The motivation for this study: Tsirelson's equation
A result of Zvonkin [23] (see also ) asserts that the stochastic differential equation driven by BM: where b(·) is only assumed to be bounded and Borel enjoys strong uniqueness. (For h such that 1 Then, the question arose whether the same strong uniqueness result might still be true with a bounded Borel drift depending more generally on the past of X, i.e.: (Uniqueness in law is ensured by Girsanov's theorem). Tsirelson gave a negative answer to this question by producing the drift: for any sequence t k ↓ 0 as k ↓ −∞.
To prove that the solution is non-strong, it suffices to study the discrete time skeleton equation: i.e.: Slight modifications of our previous arguments show that: ∀k, {η k } is independent from the BM, and uniformly distributed on [0, 1); moreover, ∀t, ∀t k ≤ t, For many further references, see Tsirelson's web page [20].
5 Some related questions and final comments a). The case C 1 [only P * µ solution] gives a beautiful example where: since in the C 1 case, j F η j = F η −∞ is trivial. This is a discussion which has been a trap for a number of very distinguished mathematicians... See, e.g., N. Wiener ([21], Chap 2) where he assumes that ∩ and ∨ may be interverted for σ-fields; if so, from Wiener's set-up, this would lead to: K-automorphism is always Bernoulli! b). As (26) above clearly shows, any solution to Tsirelson's equation (22)-(23) is not strong, i.e., (X t ) cannot be recovered from the Brownian motion (B t ) in (22)- (23).
Nevertheless, as shown in Emery-Schachermayer [7], the natural filtration of X, that is: (F X t ) t≥0 is generated by some Brownian motion (β t ) t≥0 ; thus, in that sense, the filtration (F X t ) t≥0 is a strong Brownian filtration. The question then arose naturally whether under any probability Q on C([0, 1], R), equivalent to Wiener measure, the natural filtration of the canonical coordinate process is always a strong Brownian filtration. This is definitely not the case, as shown, e.g., in Dubins-Feldman-Smorodinsky-Tsirelson [6]. The paper [7] contains a number of important references around the topic treated in [6], where again the role of B. Tsirelson has been crucial. In particular, it is another beautiful result of B. Tsirelson [19] that the natural filtration of the Brownian spider with N(≥ 3) legs is not strongly Brownian. c). Related to the problem b), we may ask the following question: For a solution (η k ) k≤0 of equation (1), when does there exist some sequence θ = (θ k ) k≤0 of independent random variables such that η k ∈ F θ k for any k ≤ 0? There are lots of studies in the case where η is a stationary process; some positive answers to this question are found in [17] and [12]. See [16] for historical remarks and related references. d). For stationary processes with common law µ on a state space S with a continuous group action of a locally compact group G, Furstenberg [10, Definition 8.1] introduced the notion of a "µ-boundary". When we confine ourselves to the case where S = G with the canonical group action and where the noise process is assumed to be identically distributed, Furstenberg's µ-boundary is essentially the same as a strong solution in our terminology, and Theorem 14.1 of [10] coincides with our Theorem 3.4.
Let G be a compact group with countable basis endowed with metric d and let S be a compact space with continuous G-action. Furstenberg ( [9]), in his study of stationary measures, proved the following: If the action is distal, i.e., for any x, y ∈ S with x = y, it holds that inf g∈G d(gx, gy) > 0, then the action is stiff, i.e., for any probability law µ on G whose support generates G, any µ-invariant probability measure on S is G-invariant. Coming back to our setting, we assume S = G with the canonical group action. Then the well-known theorem by Birkhoff [2] and Kakutani [15] shows that we may choose the metric d so that it remains invariant under the action of G, i.e., d(gx, gy) = d(x, y) for any g ∈ G. Thus the canonical action of G may be assumed to be distal, and we see that, then, the canonical action is stiff. e). We would also like to point out the relevance of the Itô-Nisio [14] study of all stationary solutions (X t ) −∞<t<∞ of some stochastic differential equations driven by Brownian motion (B t ) −∞<t<∞ . They discuss whether either of the following properties holds: s,t for s < t, where we have used obvious notations for σ-fields. Itô-Nisio [14, Section 13] identify cases where (ii) and (iii) hold, but not (i); this is similar to the situation in Tsirelson's original equation (22)- (23), or more generally the C 1 case. But, even worse, Itô-Nisio [14,Section 14] also discuss a case where (iii) does not hold. This case originates from Girsanov's equation [11] with diffusion coefficient |x| α for α < 1/2, which is well-known to generate non-uniqueness in law.
As a word of conclusion, although we do not claim that equation (1) has a deep "cosmological value", it would probably never come to the mind of "universe historians" that, in some cases, despite "the emptiness of the beginning", today's state may be independent of the evolution mechanism. Thus, seen in this light, Tsirelson's equation, and its abstraction (1) provide us with a beautiful and mind-boggling statement.