Characterization of maximal Markovian couplings for diffusion processes

Necessary conditions for the existence of a maximal Markovian coupling of diffusion processes are studied. A sufﬁcient condition described as a global symmetry of the processes is revealed to be necessary for the Brownian motion on a Riemannian homogeneous space. As a result, we ﬁnd many examples of a diffusion process which admits no maximal Markovian coupling. As an application, we ﬁnd a Markov chain which admits no maximal Markovian coupling for speciﬁed starting points.


Introduction
The concept of coupling is very useful in several areas in probability theory. Here, given two stochastic processesX (1) t andX (2) t on a common state space M , a stochastic process X t = (X (1) t , X (2) t ) on M × M defined on a probability space (Ω, , ) is called a coupling ofX if X (i) and X (i) have the same law for i = 1, 2. A characteristic of couplings in which we are interested is the coupling time T : T (X) := inf{t > 0 | X (1) s = X (2) s for any s > t}. (1.1) In many applications, we would like to make [T (X) > t] as small as possible by taking a suitable coupling. The well-known coupling inequality provides a lower bound for this probability as follows: where θ t is the shift operator (see [16] for example). We call a coupling X maximal if the equality holds in (1.2) at any t > 0. As shown in [23], a maximal coupling always exists if M is Polish and bothX (1) andX (2) are cadlag processes (for discrete time Markov chains, a maximal coupling exists on more general state spaces; see [7]).
A significance of coupling methods is emphasized when we deal with couplings of Markov processes because of their deep connection with analysis (for example, see [3; 5; 6; 10; 27] and references therein). Let ({X t } t≥0 , { x } x∈M ) be a Markov process on M . We consider the case that X is a coupling of X . It means that the law ofX (i) equals x i • X −1 for i = 1, 2 for some x 1 , In this case, many couplings appeared in application inherit a sort of Markov property from the original process. For example, the well-known Kendall-Cranston coupling (see [6; 10; 27]), which is a coupling of the Brownian motion on a complete Riemannian manifold, becomes a Markov process. Indeed, intuitively saying, we construct it by integrating a "coupling of infinitesimal motions" of two Brownian particles. In this paper, we formulate a Markovian nature of couplings in the following way: Definition 1.1. We call a coupling X = (X (1) , X (2) ) of (X , x 1 ) and (X , x 2 ) Markovian when (θ s X) · is a coupling of (X , X (1) s ) and (X , X (2) s ) under [ · | X u , 0 ≤ u ≤ s] for any s ≥ 0.
This definition means that conditioning on the past trajectories preserves the property that X is a coupling of the original Markov process X in the future. Note that X is Markovian if X itself is a Markov process on the product space M × M . Although Markovian coupling naturally appears in many cases, it is quite unclear whether Markovianity is compatible with maximality. Hence the following basic question arises; When does (or does not) a maximal Markovian coupling exist? Such a question has appeared repeatedly in various contexts in the literature. For example, K. Burdzy and W.S. Kendall [3] considered a similar problem in connection with estimates of a spectral gap (see Remark 2.4 below for the relation between maximal couplings and spectral gap estimates). It has been believed that maximal couplings are non-Markovian in general (see [7; 8; 18] for discrete case; see Remark 7.2 also).
The purpose of this paper is to give an answer to the question raised above for a class of Markov processes. Suppose that X is a diffusion process. Let us define the following property introduced in [13], which is closely related to the existence of a maximal Markovian coupling of a diffusion process. Reflection structure is a generalization of a geometric structure behind the mirror coupling of the Euclidean Brownian motion. To see it, let us suppose M = d and that X is the Brownian motion for a moment. Let R be the mirror reflection with respect to the (d − 1)-dimensional hyperplane H := {z ∈ d | |x 1 −z| = |x 2 −z|} bisecting x 1 and x 2 . Then the so-called mirror coupling (X (1) , X (2) ) is given as follows: where τ is the first hitting time of X (1) to H. Obviously, the mirror coupling is a strong Markov process as an d × d -valued process. In addition, the fact T = τ implies that the mirror coupling is maximal. We can easily verify that the mirror reflection R on d carries a reflection structure. In general, the same construction of a coupling as (1.3) still works if there exists a reflection structure with respect to (x 1 , x 2 ). We also call it the mirror coupling. We can show that the mirror coupling is a maximal Markovian coupling as well ( [13], Proposition 2.2). It means that a reflection structure implies the existence of a maximal Markovian coupling.
Our main result asserts that a reflection structure is also necessary for the existence of a maximal Markovian coupling in the following framework: To the best of the author's knowledge, such a qualitative necessary condition for the existence of a maximal Markovian coupling is not known for any Markov process until now. Moreover, this simple characterization helps us to find examples of diffusion processes which admits no maximal Markovian coupling. Actually, as we will see, there is a plenty of examples where no reflection structure exists for any pair of starting points (Theorem 6.6). Though homogeneity of the state space provides much symmetries, it is not sufficient for the existence of a reflection structure in most cases. Note that the latter part of Theorem 1.3 also asserts the uniqueness of maximal Markovian couplings. On one hand, it is shown in [13] that the mirror coupling is a unique maximal Markovian coupling when there exists a reflection structure in more general framework than Theorem 1.3 including the Brownian motion on a complete Riemannian manifold. On the other hand, Theorem 1.3 asserts the uniqueness without a priori assumption on the existence of a reflection structure though a stronger assumption is imposed on the state space.
As an application of Theorem 1.3, we obtain a finite state, discrete time Markov chain which admits no maximal Markovian coupling for specified starting points (Theorem 7.1). A characterization of maximal Markovian couplings given in Theorem 1.3 heavily depends on the continuity of sample paths. It does not seem to be so easy to establish a similar characterization for Markov chains. Thus we will take a different approach. We use Theorem 1.3 to show the claim by considering a sequence of Markov chains which approximates a diffusion process.
In the rest of this section, we state the organization of this paper. In section 2, we introduce an initial framework of our argument on the state space and the diffusion process on it. It is more general than what assumed in Theorem 1.3. In section 3, first we discuss some basic properties of maximal Markovian couplings on the framework introduced in section 2. Next we show in Proposition 3.11 that the existence of a maximal Markovian coupling carries a weak symmetry. It asserts that, at each time t ∈ [0, ∞), one particle places an antipodal point of the other particle each other with respect to a set S t ⊂ M until they meet. We call S t "mirror" in the sequel because it plays a role of {x ∈ M | Rx = x} if there is a reflection structure. It should be remarked that the mirror is a non-random set while it may depend on the time parameter t. In section 4, we derive a stronger symmetry under an additional condition (Assumption 3). There we show that Assumption 3 is a sufficient condition for the mirror to be independent of t (Proposition 4.2

Framework
In this section, we will introduce some notations and properties that are used throughout this paper. Let (M , d) be a metric space. We review some concepts on metric geometry in order to introduce additional properties on M . We call a curve γ : We assume M to be a complete, proper geodesic space that has no pair of branching geodesics. Note that all of these assumptions are satisfied if M is a connected complete Riemannian manifold or an Alexandrov space. In these cases, the nonbranching property is an easy consequence of the Toponogov triangle comparison theorem (see [2; 4], for example).
Let µ be a positive Borel measure on M satisfying 0 < µ(B) < ∞ for every metric ball B of positive radius. Note that supp[µ] = M holds. Let ({X t } t≥0 , { x } x∈M ) be a conservative diffusion process on M . We assume that there exists a strictly positive, symmetric transition density function p t (x, y) with respect to µ. That is, holds for any A ∈ (M ). In addition, we assume that p t (x, y) is jointly continuous as a function of t and y. All of these assumptions imposed on ({X t } t≥0 , { x } x∈M ) are satisfied for a broad class of symmetric diffusions including the Brownian motion on a stochastically complete, complete Riemannian manifold. In this case, µ is chosen to be the Riemannian volume measure. Note that the local parabolic Harnack inequality implies the existence and continuity of p t (see [21; 22]). For cases enjoying the inequality, see, for example, [1; 19] and references therein. To make a connection between the behavior of X t and the metric structure of M , we assume the following: There exists a decreasing sequence {t n } n∈ of positive numbers with lim n→∞ t n = 0 such that d(x, z) ≤ d( y, z) holds if p t n (x, z) ≥ p t n ( y, z) for infinitely many n ∈ . for any x, y ∈ M . This relation holds true for the Brownian motion on a Lipschitz Riemannian manifold [17]. We also state two examples having the same property. First one is a diffusion process associated with the sub-Laplacian on a nilpotent group (see [24]). The second is a canonical diffusion process on an Alexandrov space (see [15; 26]). These two cases also satisfy all other assumptions as stated above (see [14] for the latter one). For later use, we remark that the limit in (2.1) is locally uniform in x, y ∈ M in all cases mentioned above. It should be noted that the canonical diffusion process on the Sierpinski gasket enjoys Assumption 1 while it fails (2.1) (see [12], cf. [13]). But, unfortunately, it is not included in our framework because minimal geodesics on the Sierpinski gasket can branch.
In the rest of this paper, we assume the following: Given (x 1 , x 2 ) ∈ M × M \ D, a coupling X = (X (1) , X (2) ) of (X , x 1 ) and (X , x 2 ) defined on a probability space (Ω, , ) is maximal and Markovian.
The next remark concerning to Markovian couplings is essentially due to Y. Nagahata.

Remark 2.2.
The following example shows that Markovianity of couplings is strictly weaker than the Markov property as an M × M -valued process. Take two independent Brownian motions Y t and Y t on with Y 0 = 1 andŶ 0 = −1. Set We can easily verify that (Y (1) , Before closing this section, we give a remark on the coupling inequality (1.2). The right hand side of (1.2) is given as a total variation of measures on the path space. To handle it, we show that there is a simpler expression when we consider a coupling of Markov processes. Let us define ϕ t (x, y) by By definition, we have Proof. Let E ∈ (M ) be the positive part of a Hahn decomposition of For any A ∈ (C([0, ∞) → M )), the Markov property implies that In the same way, Hence the conclusion follows.
Let T = T (X) be the coupling time as defined in (1.1). By Lemma 2.3, (1.2) is the same as Thus the maximality of X implies the equality in (2.2) for any t > 0.

Remark 2.4.
In the same way as Lemma 2.3, we can express the notion of maximality based on (2.2) instead of (1.2) for couplings of any Markov process. With the aid of this formulation, maximal couplings of a Markov process are related to the spectral gap estimate as follows (cf. [3]). Suppose that µ(M ) < ∞ and p t (x, y) has the following expression: where c > 0 and λ > 0 are constants, g and R are (sufficiently regular) functions and R(t, x, y) decays faster than e −λt as t → ∞ uniformly in x, y. The Mercer theorem guarantees that it is the case if M is a compact Riemannian manifold with or without boundary and X is the (reflecting) Brownian motion. In this case, λ is the first nonzero eigenvalue of −∆/2 (with Neumann boundary condition). By the equality in (2.2), we can easily show that any maximal coupling X with It means that a maximal coupling provides an upper bound of the spectral gap by the decay rate of = λ and hence X is efficient in the sense of [3]. Note that, as the following example indicates, maximal couplings are not always efficient. Take 0 < a 1 < a 2 . Let M = M 1 × M 2 where M i is a circle of length a i with a homogeneous metric. We can easily see that there is a mirror coupling X starting from (x, y) and (x ′ , y) for any x,

Existence of a mirror
We begin with basic properties of the transition density which easily follow from our assumption. The symmetry of p t and the Schwarz inequality imply Proof. It suffices to show "only if" part. The equality in (3.1) implies p t/2 (x, z) = p t/2 ( y, z) for any z ∈ M since both of p t/2 (x, ·) and p t/2 ( y, ·) are L 1 -normalized, positive and continuous. In particular, p t/2 (x, y) = p t/2 (x, x) = p t/2 ( y, y) holds. By applying the same argument iteratively, we obtain p t/2 n (x, z) = p t/2 n ( y, z) for any z ∈ M and n ∈ .
for any bounded continuous function f . Thus, by letting n → ∞, we obtain f (x) = f ( y). Since f is arbitrary, x = y follows.
the dominated convergence theorem together with the conservativity of X implies Hence the conclusion follows.
For each s, t, u > 0, the following hold: Note that Lemma 3.2 guarantees that the above expectations are well-defined.
By the maximality of X and the definition of T , Since X is Markovian, the coupling inequality for Thus we obtain ϕ t+s (x 1 , . Take E ∈ (M ). By the definition of ϕ s , we have and hence we have By taking a supremum on since equality must hold in (3.2) -a.s. by the above argument. The equality (3.3) yields Hence (ii) follows.

Remark 3.4. The argument in the proof of Lemma 3.3 implies
where the infimum is taken on any probability measure . It means that the law of maximal Markovian coupling solves the Monge-Kantorovich problem for with the cost function ϕ s (for the Monge-Kantorovich problem, see [25] and references therein, for example).
Let us define a measure µ D t on M by We define an embedding ι : By Lemma 3.3 (i), we can show the following as in the proof of Proposition 3.5 in [13]: Note that {T ≤ t} ⊂ {X t ∈ D} obviously holds. In addition, by the maximality and Proposition 3.5, . Thus Proposition 3.5 yields the following: The following lemma asserts that µ 0,t is nondegenerate.
Proof. Suppose µ 0,t ≡ 0 for some t > 0. Then we have and hence p t (x 1 , z) = p t (x 2 , z) holds for every z ∈ M . Thus Lemma 3.1 asserts x 1 = x 2 . But it contradicts with the choice of x 1 and x 2 .
Proof. Take z ∈ H 0 (x, y). Let γ be a minimal geodesic joining x and z and γ * a minimal geodesic joining y and z. Take w on γ * with w = y, z. Then the triangle inequality asserts Since x = y and geodesics on M cannot branch, the equality cannot hold in the above inequality. Thus the fact z ∈ H 0 (x, y) implies Hence w ∈ E * 0 (x, y) holds. Since we can take w as close to z as possible, z is an accumulation point of E * 0 (x, y). By the same argument, z is also an accumulation point of E 0 (x, y). These arguments y). The converse inclusion obviously holds.
For x, y ∈ M and t > 0, let us define E t (x, y), E * t (x, y) and H t (x, y) as follows: For x, y ∈ M and t ≥ 0, let us define F t (x, y) and F * t (x, y) as follows: Recall that {t n } n∈ is given in Assumption 1. For simplicity, we denote E t (x 1 , x 2 ), F t (x 1 , x 2 ), etc. by E t , F t , etc. respectively. Note that the continuity of p t (x, y) implies E t (x, y) ⊂ F t (x, y) and E * t (x, y) ⊂ F * t (x, y). Proposition 3.9. E 0 (x, y) = F t and E * 0 (x, y) = F * t hold for µ 0,t -a.e.(x, y).
Proof. It suffices to show the former equality because the latter is shown in the same manner. First we consider the case t = 0. By Assumption 1, we have for any x, y ∈ M . Here the last equality follows from Lemma 3.8. It implies E 0 = F 0 . For t, s > 0, Lemma 3.3 (i) and the definition of ϕ s yield Since E s (x, y) ∪ H s (x, y) is the positive part of a Hahn decomposition of (p s (x, ·) − p s ( y, ·))dµ, holds for µ 0,t -a. e.(x, y). Note that H s (x, y) in (3.6) cannot be omitted because µ(E s+t ∩ H s (x, y)) may be positive. By a similar argument, we also obtain for µ 0,t -a.e.(x, y). First we observe what follows from (3.6). y)) is open and µ has a positive measure on every metric ball of positive radius, (3. for µ 0,t -a.e.(x, y). By Assumption 1, holds. Combining (3.8) with (3.9), we obtain for µ 0,t -a.e.(x, y). Next we observe what follows from (3.7). The first inclusion in (3.4) implies Here the first inclusion follows from k∈ l≥k E c t+t l ⊂ l≥n E c t+t l and the second follows from for µ 0,t -a. e.(x, y).
The following corollary will be used in the next section.
We can prove Corollary 3.10 by a similar argument as in the proof of Proposition 3.9 based on Lemma 3.3 (ii) instead of Lemma 3.3 (i).

Proof.
Take ω ∈ Ω 0 . By Lemma 3.8, (3.14) It suffices to consider the case T (X(ω)) > t. By the definition of S t , there exist sequences {z n } n∈ ⊂ F t and {z * n } n∈ ⊂ F * t such that lim n→∞ z n = lim n→∞ z * n = z. By the definition of F t and F * t , for each n ∈ , there exists a strictly increasing sequence {k n } n∈ ⊂ such that z n ∈ E t+t k n and z * n ∈ E * t+t k n holds. Since the transition density is continuous in time, there exists s n < t k n satisfying t + s n ∈ and (p t+s n (x 1 , z n ) − p t+s n (x 2 , z n )) ∧ (p t+s n (x 2 , z * n ) − p t+s n (x 1 , z * n )) > 0. It implies z n ∈ E t+s n and z * n ∈ E * t+s n . Since t + s n ∈ and ω ∈ Ω 0 , (3.13) yields E t+s n ⊂ F t+s n ⊂ E 0 (X t+s n (ω)), E * t+s n ⊂ F * t+s n ⊂ E * 0 (X t+s n (ω)).
Thus any minimal geodesic joining z n and z * n must intersect H 0 (X t+s n (ω)). Take w n from the intersection. Then we have d(z n , w n ) ∨ d(z * n , w n ) ≤ d(z n , z * n ) and hence lim n→∞ d(z, w n ) = lim n→∞ d(z n , w n ) = lim n→∞ d(z * n , w n ) = 0.

A weak characterization
At the beginning, we introduce the following additional condition: Assumption 3. p t (x, y) ≤ p t (x, x) holds for any t > 0 and x, y ∈ M . Furthermore, the equality holds if and only if x = y. Remark 4.1. The Brownian motion on a Riemannian homogeneous space satisfies Assumption 3. Indeed, for any isometry g : M → M , p t (x, y) = p t (g x, g y) holds for x, y ∈ M . Since the action of the isometry group is transitive, p t (x, x) = p t ( y, y) holds. Hence (3.1) and Lemma 3.1 yield Assumption 3. The above argument indicates that a hypoelliptic symmetric diffusion process on a homogeneous space generated by invariant vector fields also satisfies Assumption 3. A basic example is the diffusion process on a Heisenberg group associated with the sub-Laplacian. For the proof, we show the following auxiliary lemma.

Lemma 4.3. For any s, q > 0 and measurable A ⊂ M × M ,
(4.1) Proof. Note that we have

Thus (3.3) and Corollary 3.6 yield
Thus (4.1) holds. Note that, by Lemma 3.1, ϕ s (x, y) > 0 holds if and only if (x, y) / ∈ D. By virtue of Corollary 3.6, we can easily show that the support of the measure E → µ 0,q (E × M ) equals E q . Thus the conclusion follows.

Proof of Proposition 4.2.
We may assume t > t 1 without loss of generality. Take q, s > 0. Note that Lemma 4.3 implies [X q ∈ D, T > s + q] = 0. If X (1) q / ∈ E s (X q ), then we have and hence X (1) q = X (2) q holds by Assumption 3. The same argument also works for X (2) q and E * s (X q ) instead of X (1) q and E s (X q ). Thus X (1) q ∈ E s (X q ) and X (2) q ∈ E * s (X q ) hold on {T > s + q} -a.s.. Therefore Proposition 3.9 and Corollary 3.10 yield By applying this inclusion in the case (q, s) = (u + t n , t − t n ), and hence F u ⊂ F t+u . By the same argument, we obtain F * u ⊂ F * t+u . Thus S u ⊂ S t+u holds. In order to show S t+u = S u , suppose S t+u \ S u = and take z ∈ S t+u \ S u . Then either z / ∈ F u or z / ∈ F * u holds. We only deal with the case z / ∈ F u because the other one will be treated in the same way. Take δ > 0 so small that B δ (z) ∩ F u = holds. For q > 0, we can take (x, y) ∈ E q × E * q so that it satisfies Note that such a pair (x, y) exists by Proposition 3.9 and Lemma 3.7. The expression (4.3) in the case q = u yields F u c ⊂ F * u and hence B δ (z) ⊂ F * u holds. It implies Take (x, y) ∈ E t+u × E * t+u so that it satisfies (4.3) in the case q = t + u. Then Lemma 3.8 yields S t+u = H 0 (x, y). Since z ∈ S t+u , there is a sequence {z n } n∈ in E 0 (x, y) such that z n converges to z. Then clearly z n / ∈ E * 0 (x, y) = F * t+u for any n ∈ , but it contradicts with (4.4). Thus we obtain S t+u = S u .
In what follows, we will prove S u = H 0 . By definition, We turn to the converse inclusion. Assumption 1 guarantees that x 1 ∈ E t n and x 2 ∈ E * t n hold for sufficiently large n. Take such n and (x, y) ∈ E t n × E * t n so that it satisfies (4.3) in the case q = t n . Then, Lemma 3.8 yields y). y). Suppose H 0 \ S t n = and take w ∈ H 0 \ S t n . Take a minimal geodesic γ joining x 1 and w and γ ′ joining w and x 2 . We define a pathγ by concatenating γ and γ ′ at w. Then, the discussion in the proof of Lemma 3.8 implies Here we identify each geodesic with the set of its trajectory. Since H 0 (x, y) = S t n ⊂ H 0 , we obtaiñ γ ∩ H 0 (x, y) = . It contradicts with the fact that the endpoints x 1 and x 2 ofγ belong to E 0 (x, y) and E * 0 (x, y) respectively. Hence H 0 = S t n = S u follows.

Remark 4.4.
The mirror S t may depend on time parameter t in general. To see it, we observe the following simple example. Take x 1 , x 2 ∈ d with x 1 = x 2 and v ∈ d . Set H := {z ∈ d | |x 1 − z| = |x 2 − z|} and S t := t v + H. Let R t be the mirror reflection with respect to S t . Let us define two process Y where B t is the standard Brownian motion on d and τ : We can easily verify that (Y (1) , Y (2) ) is a maximal Markovian coupling of two Brownian motions with the drift v. Strictly speaking, this is not the case because the symmetry of p t fails. The author does not know that such a example exists in the class of symmetric diffusions.
The following theorem provides a weak characterization of maximal Markovian couplings.
t ) is written as follows -almost surely: Before proving Theorem 4.5, we show the following auxiliary lemma. then (x, y) Proof. Suppose that y ∈ M \ {x} satisfies (4.7). If x ∈ H 0 , then (4.7) obviously fails when z = x. Thus the cases x ∈ H 0 and y ∈ H 0 are excluded. Suppose x, y ∈ E 0 . Let γ be a minimal geodesic joining x 2 and x. Take z 0 ∈ γ ∩ H 0 . Then we have By the same argument, d( y, . Since x = y, we can take a minimal geodesic joining x 2 and y that branches from γ at z 0 . It contradicts with our assumption. In the same way, we can exclude the case x, y ∈ E * 0 . Hence the former assertion follows. Let us turn to the latter assertion. We consider the case that (4.7) holds for (x, y) = (x ′ , y ′ ) and (x, y) = (x ′ , y ′′ ) for x ′ ∈ M and y ′ , y ′′ ∈ M \ {x ′ }. Then the former assertion implies ( y ′ , y ′′ ) ∈ E 0 × E 0 ∪ E * 0 × E * 0 . Since (4.7) holds for (x, y) = ( y ′ , y ′′ ), we obtain y ′ = y ′′ by using the former assertion again.

Proof of Theorem 4.5. Let us define a set A ⊂ M as follows:
A := x ∈ M | there exists y ∈ M \ {x} such that (4.7) holds .
For x ∈ A, we define Rx := y, where y is a point satisfying (4.7). Lemma 4.6 guarantees that R is well-defined. For x ∈ H 0 , we define Rx := x. SetÂ = A ∪ H 0 . First we show thatÂ is closed and that R is continuous onÂ. Let {x n } n∈ be a sequence inÂ that converges to x ∈ M . Take z 0 ∈ H 0 . Since d(z 0 , x n ) = d(z 0 , Rx n ) holds for any n ∈ , {d(z 0 , Rx n )} n∈ is bounded. Thus the properness of M yields that {Rx n } n∈ has an accumulation point y. Note that y = x holds if x ∈ E 0 ∪ E * 0 . Indeed, if x ∈ E 0 , then x n ∈ E 0 for sufficiently large n and Lemma 4.6 implies Rx n ∈ E * 0 for such n. Choose a subsequence {Rx n k } k∈ that converges to y. Then we have holds for any z ∈ H 0 . Thus x ∈Â and y = Rx. Since the choice of an accumulation point of {Rx n } n∈ is arbitrary, the above argument also implies the continuity of R onÂ.
Note that Theorem 4.5 implies that X is a strong Markov process on M × M . Thus Remark 3.13 together with Proposition 4.2 yields the following:

Riemannian homogeneous spaces
In this section, we derive a stronger characterization of maximal Markovian couplings under the following assumptions: Suppose that there is a maximal Markovian coupling X of (X , x 1 ) and (X , x 2 ). Then there exists a reflection structure R with respect to (x 1 , x 2 ). Furthermore, X is the mirror coupling determined by R. In the rest of this section, we use the notation in Theorem 4.5. To complete the proof of Theorem 5.1, it suffices to show that the reflection R makes the process invariant under Assumption 4 and Assumption 5.

Lemma 5.3. Under Assumption 4, R is an isometry on M .
Proof. Since Rx = x holds for x ∈ H 0 , d(x, y) = d(Rx, R y) trivially holds for x, y ∈ H 0 . When x / ∈ H 0 and y ∈ H 0 , d(x, y) = d(Rx, y) = d(Rx, R y) follows directly from the definition of R. For x ∈ E 0 and y ∈ E * 0 , we have since every curve joining x and y must intersect H 0 . Finally we consider the case x, y ∈ E 0 . Take s > 0 so small that x ∈ E s and Rx ∈ E * s . Take δ > 0 so small that Thus, the strong Markov property, Theorem 4.5, Corollary 4.7 and Corollary 3.6 yield In the same way, we have Now we claim that, if z, w ∈ E 0 or z, w ∈ E * 0 , Let γ be a minimal geodesic joining w and Rz and take If the equality holds in the above inequality, then we can take a minimal geodesic joining w and z that branches from γ at z 0 . Hence the claim follows.
Hence (5.7) also follows as we did after (5.6) had been obtained.
Finally we consider the case x ∈ E 0 and y ∈ E 0 ∪ E * 0 . Take s > 0 so small that x ∈ E s . Take δ > 0 sufficiently small. Now we have By Theorem 4.5 (ii) and Lemma 5.3, In a similar way as in (5.6), s+t ∈ B δ (R y), s ≤ T < s + t . (5.10) By replacing X (1) , x and y with X (2) , Rx and R y in (5.8), we obtain a corresponding decomposition. Combining it and (5.8) with (5.9) and (5.10), we obtain Here we have Thus, dividing both side of (5.11) by µ(B δ (x))µ(B δ ( y)) and tending δ to 0, we obtain T , Rx) p t (Rx, R y). (5.12) Note that Corollary 3.6 implies By the same argument, we have Here the last equality follows from (5.7). By substituting (5.13) and (5.14) into (5.12), the desired result follows.

Examples: Riemannian symmetric spaces
In this section, we consider some examples of the Brownian motion on a Riemannian symmetric spaces. Since any Riemannian symmetric space is homogeneous, we can apply Theorem 1.3. Thus a maximal Markovian coupling exists if and only if there is a reflection structure. The following three examples indicate that the existence of a reflection structure imposes a strong restriction on the underlying space. d M denotes the distance function on a metric space M .
First we review the cases that M is simply connected and has a constant curvature, That is, M is either a sphere d , a Euclidean space d or a hyperbolic space d corresponding to the signature of the curvature. As studied in Example 4.6 in [13], there is a reflection structure with respect to (x 1 , x 2 ) for any (x 1 , x 2 ) ∈ M × M \ D.  Note that an involutive isometry whose fixed points form a submanifold of codimension 1 exists if and only if M is of constant curvature (see [9]). Now suppose that there exists a reflection structure on M . The induced map R is an involutive isometry. Moreover, the fixed points H 0 must be of codimension 1 since H 0 separates M into two disjoint open sets. Thus, if M has a non-constant curvature, then there is no reflection structure with respect to (x 1 , x 2 ) for any x 1 , x 2 ∈ M . for some y 1 , y 2 ∈ \ {0} without loss of generality. For simplicity, we assume y 2 1 + y 2 2 = 1. For In the same way, we obtain cos d d (x 2 , [z 1 : · · · : z d+1 ]) = | y 1 z 1 − y 2 z 2 |.
Note that H 0 is not a manifold since it has a singularity at [0 : 0 : z 3 : · · · : z d+1 ]. Thus there is no reflection structure by Remark 6.2.
Example 6.5. (Tori) Let us consider the d-dimensional torus d for d ≥ 2. Here = / . We endow with a flat metric induced from and d the product metric. Take (x 1 , Let us denote them by x 1 = (x 11 , . . . , x 1d ), x 2 = (x 21 , . . . , x 2d ) for x i j ∈ . We claim that there exists a reflection structure if and only if there is k ∈ {1, . . . , d} such that x 1 j = x 2 j for any j = k.
First we show the "if" part. For simplicity, we assume k = 1. Then we can easily verify that a map R defined by R( y 1 , . . . , y d ) = (x 1 + x 2 − y 1 , y 2 , . . . , y d ) carries a reflection structure with respect to (x 1 , x 2 ). Next we show the "only if" part. It suffices to show that, for each j 1 , j 2 ∈ {1, . . . , d} with j 1 = j 2 , x 1 j 1 = x 2 j 1 or x 1 j 2 = x 2 j 2 must hold. By symmetry, we may assume ( j 1 , j 2 ) = (1, 2) without loss of generality. For a subset M 0 ⊂ M , we endow M 0 with the geodesic metric inherited from M . It means that, for x, y ∈ M 0 , d M 0 (x, y) is the infimum of the length of all rectifiable curve joining x and y in M 0 . For k = 3, . . . , d, take z k ∈ so that d (x 1k , z k ) = d (x 2k , z k ) holds. SetĤ := ( y 1 , . . . , y d ) ∈ d y k = z k for k = 3, . . . , d . Note thatĤ where λ is the first nonzero eigenvalue of the Markov chain. As observed in Remark 2.4, any maximal coupling satisfies (2.3). Thus no maximal coupling can be a Markov process on the product space.
For the proof of Theorem 7.1, we construct an approximating sequence of couplings W (m) of Markov chains that converges in law to a coupling of two Brownian motions on d . Let {Z n,i } n∈ , i∈{1,...,d} be -valued, independent and identically distributed random variables defined by . We show the following auxiliary lemma which asserts the local central limit theorem on . Lemma 7.3 seems to follow easily from the local central limit theorem forp n (x, y), but we need to estimate that fluctuations are so small as to be negligible. Our proof is based on the arguments in Chapter 2 of [20]. Though such an extension may be well-known, we will give a proof for completeness. Here the first equality follows from the fact | m ′ l=1 Z l,1 | ≤ m ′ . We decompose the right hand side of (7.2) as follows: