Rigidity for Markovian maximal couplings of elliptic diffusions
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Abstract
Maximal couplings are (probabilistic) couplings of Markov processes such that the tail probabilities of the coupling time attain the total variation lower bound (Aldous bound) uniformly for all time. Markovian (or immersion) couplings are couplings defined by strategies where neither process is allowed to look into the future of the other before making the next transition. Markovian couplings are typically easier to construct and analyze than general couplings, and play an important role in many branches of probability and analysis. Hsu and Sturm, in a preprint circulating in 2007, but later published in 2013, proved that the reflectioncoupling of Brownian motion is the unique Markovian maximal coupling (MMC) of Brownian motions starting from two different points. Later, Kuwada (Electron J Probab 14(25), 633–662, 2009) proved that the existence of a MMC for Brownian motions on a Riemannian manifold enforces existence of a reflection structure on the manifold. In this work, we investigate suitably regular elliptic diffusions on manifolds, and show how consideration of the diffusion geometry (including dimension of the isometry group and flows of isometries) is fundamental in classification of the space and the generator of the diffusion for which an MMC exists, especially when the MMC also holds under local perturbations of the starting points for the coupled diffusions. We also describe such diffusions in terms of Killing vectorfields (generators of isometry groups) and dilation vectorfields (generators of scaling symmetry groups). This permits a complete characterization of those possible manifolds and their diffusions for which there exists a MMC under local perturbations of the starting points of the coupled diffusions. For example, in the timehomogeneous case it is shown that the only possible manifolds that may arise are Euclidean space, hyperbolic space and the hypersphere. Moreover the permissible drifts can then derive only from rotation isometries of these spaces (and dilations, in the Euclidean case). In this sense, a geometric rigidity phenomenon holds good.
Keywords
Characteristic operator Coupling Diffusiongeodesic completeness Diffusion geometry Diffusion matrix Elliptic diffusion Global isometry group Homogeneous space Immersion coupling Infinitesimal generator Involutive isometry Killing vectorfield Laplace–Beltrami operator Large deviations Local perturbation condition Markovian coupling Maximal coupling Maximally symmetric space Orthonormal frame bundle Riemannian manifold Reflection coupling Rigidity Stochastic completeness Stochastic differential equation Stochastic parallel transport Stratonovich differential Strong maximum principle Topogonov comparison theorem Totally geodesic submanifoldMathematics Subject Classification
60G05 58J65 60J601 Introduction
The major drawback of all these constructions is they are typically very implicit; in most cases, it is extremely hard, if not impossible, to make detailed calculations for such couplings. This is a strong motivation for considering Markovian couplings, which we now describe.
A natural and immediate question is, when can a maximal coupling of two diffusions be Markovian? The standard (and elegant) example in the literature is the reflectioncoupling of Euclidean Brownian motions starting from two different points: the second Brownian path is obtained from the first by reflecting the first path on the hyperplane bisecting the line joining the starting points until the first path (equivalently, the second, reflected, path) hits this hyperplane. Both paths then evolve together (“synchronously”) as a single Brownian path. Straightforward calculations, based on the reflection principle, show that this construction is in fact a Markovian maximal coupling (MMC). Furthermore, [18] proved that this is the unique such coupling for Euclidean Brownian motion. A few other examples are discussed in the literature: Ornstein Uhlenbeck processes [9], also Brownian motion on manifolds which possess certain reflection symmetries. The reflection coupling idea manifests itself throughout the area of probabilistic coupling: for example it has a natural generalization to Brownian motion on Riemannian manifolds [10, 19], involving stochastic parallel transport and development, and not requiring any symmetries of the manifold. However it seems unlikely that such generalizations will normally provide maximal couplings. Kuwada [25] investigated this question for Brownian motion on manifolds (and their generalisations to metric spaces). Under suitable mild regularity assumptions he showed that a reflection symmetry of the space is necessary for the existence of a Markovian maximal coupling of two Brownian motions started from a specified pair of points. Working under some further assumptions, he proved that the fixed point set of the symmetry (the “mirror”, characterizing this isometry) does not change with time; the maximal coupling is given simply by reflecting one process onto the other using the reflection symmetry defined by this mirror.
The aim of this paper is to develop the results of Kuwada to the case of general regular elliptic diffusions with smooth coefficients. It will be shown that Markovian maximal couplings are rare, in the sense that a stable local existence result enforces extreme global symmetry on the manifold: a kind of rigidity result. Section 2 considers implications of existence of Markovian maximal couplings for \(d\)dimensional Euclidean diffusions (“Euclidean” here meaning that the diffusion matrix is the identity matrix), under rather general regularity assumptions on the (possibly timeinhomogeneous) drift. Extending Kuwada’s argument, the existence of an MMC implies there is a mirror symmetry between the coupled processes at any given time. However the influence of the nonzero drift now means that the mirror can vary deterministically with time, making the coupled dynamics considerably more complicated. We study the evolution of the mirror in time using stochastic calculus and we obtain a functional equation that the drift must satisfy for a Markovian maximal coupling to exist. This equation can be used to characterise all timeinhomogeneous diffusions which admit such couplings.
In the timehomogeneous case the characterization can be refined under the additional hypothesis that there is also a Markovian maximal coupling under local perturbation of the starting points, which is to say, Markovian maximal couplings exist locally in a stable sense:
Definition 1
(Local Perturbation Condition (LPC)) There is \(r>0\), and initial points \(\mathbf {x}_0\) and \(\mathbf {y}_0\), such that there exists a Markovian maximal coupling of the diffusion processes X and Y starting from \(\mathbf {x}\) and \(\mathbf {y}\) for every \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0,r)\) and \(\mathbf {y}\in \mathcal {B}(\mathbf {y}_0,r)\), where \(\mathcal {B}(\mathbf {x}_0,r)\) is the open metric ball centred at \(\mathbf {x}_0\) and of radius \(r\).
We will show that, for any dimension \(d\ge 1\), LPC holds for a suitably regular Euclidean diffusion with timehomogenous drift if and only if the drift takes the form \(\mathbf {b}(\mathbf {x})=\lambda \mathbf {x}+ T\mathbf {x}+ \mathbf {c}\), where \(\lambda \) is a scalar, T is a skewsymmetric matrix and \(\mathbf {c}\) is a fixed vector. This implies that Brownian motion with constant drift and Ornstein–Uhlenbeck process are the only onedimensional examples of timehomogeneous diffusions for which there are successful Markovian maximal couplings from arbitrary pairs of starting points. In higher dimensions, for regular Euclidean diffusions under LPC, essentially the same is true except that the drift may also include a rotational component. In one dimension, even without LPC, it turns out that a Markovian maximal coupling exists between two copies of a regular diffusion started from \(x_0\) and \(y_0\) if and only if the drift is either affine or an odd function around the midpoint of the starting points.
Section 3 considers Markovian maximal couplings of Brownian motion with timehomogeneous drift on a complete Riemannian manifold M under LPC. This is the natural generalization of the context of Sect. 2, since a regular elliptic diffusion on Euclidean space furnishes the space with a Riemannian metric by means of inverting the diffusion matrix, and then the diffusion is converted into a Brownian motion with drift on the resulting Riemannian manifold, so that the Riemannian geometry serves to classify a variety of diffusions (compare the rather similar role of Fisher information in theoretical statistics). We assume that the elliptic diffusion is stochastically complete, and also diffusiongeodesically complete, in the sense that the diffusion Riemannian geometry is geodesically complete. Strikingly, LPCthen produces a geometric rigidity phenomenon, namely a complete classification of the space M as one of the three model spaces \(\mathbb {R}^d\) (Euclidean space), \(\mathbb {S}^d\) (Sphere) and \(\mathbb {H}^d\) (Hyperbolic space) depending upon the sign of the (necessarily constant) curvature K (see Theorem 38 in Sect. 3). The Euclidean case is fully covered in Sect. 2, and delivers the necessary ideas and techniques which we generalise to the manifold setup in Sect. 3 to study Markovian maximal couplings on the other two spaces. It turns out that the only drifts which can yield Markovian maximal couplings are given by the Killing vectorfields, defined as infinitesimal generators for the rigid motion group (namely, generators of oneparameter subgroups of isometries).
In this paper we confine our considerations to the case of elliptic diffusions, where there is a strong connection to Riemannian geometry, and pathcontinuity permits the formation of interfaces of codimension \(1\) separating pairs of initial points. Possible extensions to hypoelliptic diffusions or to general Markov chains are potentially of great interest, but we leave these questions as topics for future work.
1.1 Markovian maximal couplings: general properties
We complete this introduction by defining some general notation and by describing some basic general properties of Markovian maximal couplings for general Markov processes on a metric space \((M,{\text {dist}})\). Kuwada [25] derived results similar to Lemmas 2 and 3 below. For the sake of clearer exposition, and as we are primarily interested in diffusion processes, we will state the results for continuoustime Markov processes. Denote the Markov process under consideration by X.
We assume that the metric space supports a positive Borel measure m with \(0<m(B)<\infty \) for any metric ball B of finite radius. Consequently, the closed support of m is the whole of M. We further assume that for any \(t>s \ge 0 \), the conditional distribution law \(\mathcal {L}\left( X_t \mid X_s=x\right) \) is absolutely continuous with respect to m and has a probability kernel density given by \(p(s,\mathbf {x};t,\mathbf {z})\) for \(\mathbf {x}\), \(\mathbf {z}\in M\) and \(0\le s<t\).
We will be dealing with Markov processes which are possibly timeinhomogeneous, so we say a Markov process starts from \((t,\mathbf {x})\) if we are looking at the distribution law \(\mathcal {L}\left( \theta _tX \mid X_t=\mathbf {x}\right) \), where \(\theta \) denotes the timeshift operator given by \((\theta _tX)_s=X_{t+s}\).
Lemma 2
Proof
\(\square \)
Only maximality was required for Lemma 2. If in addition \(\mu \) is Markovian, then the conditional law \(\mathcal {L}\left( \theta _sX, \theta _sY \mid \mathcal {F}_s\right) \) describes a Markovian coupling of two copies of our Markov process starting from \(((s,X_s), (s,Y_s))\). Such a coupling therefore satisfies the following flow property:
Lemma 3
If \(\mu \) is a Markovian maximal coupling and \(\mu _s=\mathcal {L}\left( X_s, Y_s\right) \) then, for \(\mu _s\)almost every \((\mathbf {x},\mathbf {y})\) with \(\mathbf {x}\ne \mathbf {y}\) the conditional law \(\mathcal {L}\left( \theta _sX, \theta _sY \mid X_s=\mathbf {x}, Y_s=\mathbf {y}\right) \) gives a Markovian maximal coupling of (X, Y) starting from \(((s,\mathbf {x}),(s,\mathbf {y}))\).
Proof
This follows immediately from the maximality of \(\mu \) and the fact that \(\mu \) is Markovian. \(\square \)
Lemma 4
For each \(t \ge 0\), let \(F_t:(\Omega _1,\mathcal {F}_1) \rightarrow (\Omega _2,\mathcal {F}_2)\) be a bijective mapping between two measurable spaces such that \(F_t, F_t^{1}\) are measurable. Then, for any Markov process \(\{X_t: t \ge 0\}\) on \(\Omega _1\), \(\{F_t(X_t): t \ge 0\}\) defines a Markov process on \(\Omega _2\). Furthermore \(\{(X_t,Y_t): t \ge 0\}\) is a (Markovian) maximal coupling of Markov processes on \(\Omega _1\) if and only if \(\{(F_t(X_t),F_t(Y_t)): t \ge 0\}\) is a (Markovian) maximal coupling on \(\Omega _2\).
Proof
The first assertion is a direct consequence of the general definition of conditional expectation. The second assertion follows from the definition of maximality. \(\square \)
2 Markovian maximal couplings on Euclidean spaces
 (A1)
The drift vectorfield \(\mathbf {b}: [0,\infty ) \times \mathbb {R}^d\rightarrow \mathbb {R}\) is continuously differentiable in the second (space) variable, moreover \(\mathbf {b}\) and all its firstorder spatial partial derivatives \(\partial _i\mathbf {b}\) are bounded on compact subsets of \([0,\infty ) \times \mathbb {R}^d\).
 (A2)
For every \(t>s\ge 0\), and \(\mathbf {x},\mathbf {z}\in \mathbb {R}^d\), the conditional distribution law \(\mathcal {L}\left( X_t \mid X_s=x\right) \) is the law of a diffusion with transition probability density kernel \(p(s,\mathbf {x};t,\mathbf {z})\) (density with respect to Lebesgue measure), which is jointly continuous in all its arguments. Moreover, \(p(s,\cdot ;\cdot , \cdot )\) is positive everywhere when \(s>0\). Finally, the density \(p(s,\mathbf {x};\cdot ,\cdot ): \mathbb {R}^+ \times \mathbb {R}^d\rightarrow \mathbb {R}\) is continuously differentiable in the time variable (first unspecified variable) and twice continuously differentiable in the space variable (second unspecified variable).
Remark 5
Note that Assumption (A2) implies that the diffusion does not explode in finite time (otherwise \(p(s,\mathbf {x};t,\cdot )\) would determine a subprobability density). A sufficient condition for nonexplosion is to require that \(\mathbf {b}\) is locally Lipschitz in the space variable \(\mathbf {x}\) (which follows from Assumption (A1)) and moreover that there exists a constant C such that \(b(t,\mathbf {x}) \le C(1+t +\mathbf {x})\) for all \((t,\mathbf {x}) \in [0,\infty ) \times \mathbb {R}^d\) [17, Proposition 1.1.11]. Furthermore, the fact that \(\mathbf {b}\) is locally Lipschitz in \(\mathbf {x}\) implies the existence of a unique strong solution to the SDE corresponding to (7) for any given driving Brownian motion B [17, Theorem 1.1.8].
We will sometimes say \(\mathbf {b}\) satisfies Assumptions (A1) and (A2) if \(\mathbf {b}\) satisfies (A1) and the corresponding diffusion (whose law is unique by the above remark) has transition probability densities satisfying (A2).
Recall that we say a diffusion starts from \((t,\mathbf {x})\) if we are looking at the law \(\mathcal {L}\left( \theta _tX \mid X_t=\mathbf {x}\right) \), where \(\theta \) denotes the timeshift operator given by \((\theta _tX)_s=X_{t+s}\). The resulting process is a diffusion with the identity diffusion matrix but using timeshifted drift \(b(t+\cdot ,\cdot )\) and starting from x at time 0.
Let X and Y be two copies of this diffusion starting from \(\mathbf {x}_0\) and \(\mathbf {y}_0\) respectively.
Remark 6
The function \((s,\mathbf {x}) \mapsto p(0,\mathbf {x}_0;ts,\mathbf {x})\) satisfies a backward parabolic equation. Therefore uniqueness theory for such equations yields that there does not exist any \(s>0\) such that \(p(0,\mathbf {x}_0; s,\mathbf {z})=p(0,\mathbf {y}_0; s,\mathbf {z}) \text { for all } \mathbf {z}\in \mathbb {R}^d\). This, along with (6), implies that, for every \(s>0\), \(\mu (\tau >s)>0\) and thus \(\mu (\mathcal {M}(\mu _s))>0\). In particular, \(\mathcal {M}(\mu _s)\) is nonempty for each \(s>0\).
2.1 Coupling and the interface
Here, we show that the existence of a Markovian maximal coupling for X and Y implies that for each time t, the interface \(I(\mathbf {x}_0,\mathbf {y}_0,t)\) will be a hyperplane bisecting the straight line joining \(X_t\) and \(Y_t\).
We begin with some preparatory lemmas. Note that Brownian motion has fluctuations which are of order \(O(\sqrt{t})\) while fluctuations resulting from the drift are of order O(t). Thus, on small time scales, the Brownian behaviour should dominate. The following lemma substantiates this intuition.
Lemma 7
Proof
Let \(I=\sup \{\mathbf {y}\mathbf {x}_0: \mathbf {y}\in \mathcal {B}(\mathbf {z},\delta )\}\) and choose \(N>d\times I+1\). By continuity of \(\mathbf {b}\), there is a finite M for which \(\mathbf {b}(t,\mathbf {y}) \le M\) for all \((t,\mathbf {y}) \in [0,1] \times \mathcal {B}(\mathbf {x}_0, N)\).
Remark 8
The above lemma can be regarded as a weak form of a large deviation principle (LDP) for the diffusion X, specialized to a particular set \(B(\mathbf {z},\delta )\). The general form of the LDP can be shown to hold under the additional assumption of linear growth of the drift vectorfield, which is used to control the moments of the Radon–Nikodym derivative of the law of X with respect to that of B obtained by the Girsanov Theorem [40].
Lemma 9
It is now possible to state and prove the main result of this section, which can be seen as a stronger version of [25, Proposition 3.9], although our proof is quite different and slightly shorter.
Theorem 10
Proof
By continuity of \(\alpha (s,\cdot )\), it suffices to prove that \(H^(\mathbf {x},\mathbf {y}) \subseteq I^{}(\mathbf {x}_0,\mathbf {y}_0,s)\) and \(H^+(\mathbf {x},\mathbf {y}) \subseteq I^{+}(\mathbf {x}_0,\mathbf {y}_0,s)\).
We will first show that \(\alpha (s,\mathbf {z}^*) \ge 0\) for all \(\mathbf {z}^* \in H^(\mathbf {x},\mathbf {y})\). Suppose, in contradiction, that \(\alpha (s,\mathbf {z}^*)<0\) for some \(\mathbf {z}^* \in H^(\mathbf {x},\mathbf {y})\).
Remark 11
The above theorem shows that for a Markovian maximal coupling, for any time s, the locus \(I(\mathbf {x}_0, \mathbf {y}_0,s)\) can be viewed as a (possibly timevarying) mirror which realizes the coupling in a very explicit way, using a (possibly timevarying) reflection isometry.
The following corollary to the above lemma shows that the coupling time \(\tau \) is, in fact, the hitting time of the deterministic spacetime set \(\{(s,I(\mathbf {x}_0,\mathbf {y}_0,s)) : s>0\}\) by the process \(((s,X_s):s\ge 0)\) (equivalently, \(((s,Y_s):s>0)\)). In particular, X and Y will couple at the first time they meet. Furthermore, the interface representation described in Theorem 10 will hold almost surely for all time before coupling occurs.
Corollary 12
Proof
2.2 Time evolution of the mirror
We now analyze the timeevolution of the mirror. From Theorem 10, it follows that the mirror \(I(\mathbf {x}_0,\mathbf {y}_0,t)\) is a hyperplane for each \(t>0\). We parametrize this hyperplane by its signed distance from the origin, say l(t), together with the normal vector to the hyperplane, say \(\mathbf {n}(t)\). There is an ambiguity of sign in the choice of \(\mathbf {n}(t)\); however the next lemma states that \(\mathbf {n}(t)\) can be chosen to make this parametrization continuous up to the coupling time \(\tau \).
Lemma 13
Suppose that a Markovian maximal coupling exists for X and Y. Then there exists a continuous parametrization \(\left( (l(t), \mathbf {n}(t)): t\in [0,\tau )\right) \) of \(I(\mathbf {x}_0,\mathbf {y}_0,\cdot )\).
Proof
In fact the parametrization is not simply continuous but is also continuously differentiable:
Lemma 14
Suppose that a Markovian maximal coupling exists for X and Y. Then the parametrization \((l(t), \mathbf {n}(t))\) of the mirror \(I(\mathbf {x}_0,\mathbf {y}_0,t)\) (defined for \(t \in [0,\tau )\)) is continuously differentiable in \(t\).
Proof
Now, take any \(t_0 \in [0,\tau )\). Let \(n_i\) denote the ith component of \(\mathbf {n}\). As \(\mathbf {n}(t_0)=1\), there is an i such that \(n_i(t) \ne 0\) in a neighbourhood V of \(t_0\). The continuous differentiability of \(\displaystyle {t \mapsto ({\mathbb {I}}2\mathbf {n}(t)\mathbf {n}^\top (t))}\) implies \(n_in_j\) is continuously differentiable in V for all \(1 \le j \le d\). This implies \(n_j\) is continuously differentiable in V for all j. Differentiability of \(\displaystyle {t \mapsto l(t)\mathbf {n}(t)}\) then shows that l is continuously differentiable on V. This proves the lemma. \(\square \)
2.3 Structure of the coupling
All the tools having been assembled, it is now possible to present a rather explicit description of drifts \(\mathbf {b}\) which permit the existence of a Markovian maximal coupling of two copies \(X\) and \(Y\) of a Euclidean diffusion with the required regularity conditions.
We begin with a notational remark. For any \(\mathbf {x}\in \mathbb {R}^d\) and any hyperplane \(\underline{h}\), we denote by \(\underline{h}\mathbf {x}\) the reflection of \(\mathbf {x}\) in \(\underline{h}\). We write \(\underline{h}_k\) for the hyperplane \(\{x_k=0\}\).
The first lemma of this subsection concerns an observation concerning rotations and shifts of these Euclidean diffusions.
Lemma 15
Proof
The result follows by direct calculation using Itô calculus. \(\square \)
Remark 16
Note that the transformed drift given by (26) satisfies the regularity Assumptions (A1) and (A2). (A1) follows via the explicit form of (26) from the fact that \(\mathbf {b}\) satisfies (A1) and Q and l are continuously differentiable. (A2) for the new process \(\widetilde{X}\) follows from (24) and the fact that X satisfies (A2).
The following theorem describes Markovian maximal couplings for the class of timenonhomogeneous Euclidean diffusions satisfying suitable regularity conditions. The intuitive content of the theorem is, given an MMC (X, Y), applying deterministic timevarying rotations and translations to the ambient Euclidean space reduces this MMC to a reflection coupling in a fixed hyperplane. Thus, in a certain sense, reflection coupling is the only type of Markovian coupling that can possibly preserve maximality.
Theorem 17
 (i)Suppose the following holds for every \(x \in \mathbb {R}^d\), for the fixed hyperplane \(\underline{h}_1=\{x_1=0\}\).Then, for \(\tau _0=\inf \{t\ge 0: X_t \in \underline{h}_1\}\), the reflectioncoupling$$\begin{aligned} \mathbf {b}(t,\underline{h}_1\mathbf {x})=\underline{h}_1\mathbf {b}(t,\mathbf {x}) \end{aligned}$$(28)gives a Markovian maximal coupling between two copies of the diffusion starting from \(\mathbf {x}_0\) and \(\underline{h}_1\mathbf {x}_0\) respectively.$$\begin{aligned} Y_t= & {} {\left\{ \begin{array}{ll} \underline{h}_1X_t &{} \hbox { if }\quad t< \tau _0 \\ X_t &{} \hbox { if }\quad t \ge \tau _0 \end{array}\right. } \end{aligned}$$(29)
 (ii)Let \(Y\) be a coupled copy of \(X\). Then (X, Y) is a Markovian maximal coupling up to the maximal coupling time \(\tau \) if and only if there exist \(C^1\) curves \(Q:[0,\tau ) \rightarrow \mathbf {O}(d)\) and \(l:[0,\tau ) \rightarrow \mathbb {R}\) (compare Lemma 15) with \({Q(0)\tfrac{\mathbf {x}_0\mathbf {y}_0}{\mathbf {x}_0\mathbf {y}_0}=\mathbf {e}_1}\) and \({l(0)=\tfrac{\mathbf {x}_0^2\mathbf {y}_0^2}{2\mathbf {x}_0\mathbf {y}_0}}\), such that \((\widetilde{X}, \widetilde{Y})\) obtained from (X, Y) using the transformation (24) are reflectioncoupled according to the recipe (29). In particular, the transformed timevarying drift \(\widetilde{\mathbf {b}}\) given by (26) must satisfy$$\begin{aligned} \widetilde{\mathbf {b}}(t,\underline{h}_1\mathbf {x})=\underline{h}_1\widetilde{\mathbf {b}}(t,\mathbf {x}). \end{aligned}$$(30)
Proof
 (i)
Equation (28) implies that the process \((\underline{h}_1X_t\;:\;t \ge 0)\) has the same law as the diffusion starting from \(\underline{h}_1\mathbf {x}_0\) and thus, the reflectioncoupling (29) gives a valid coupling. Reflection in the hyperplane \(\underline{h}_1\) thus gives a reflection structure in the sense of [24, Definition 2.1]. Maximality follows from [24, Proposition 2.2].
 (ii)First, note that if \(\widetilde{X}\) and \(\widetilde{Y}\) are reflectioncoupled according to (29), then analysis of generators of \(\underline{h}_1\widetilde{X}_t\) and \(\widetilde{Y}_t\) yields (30). Now, applying part (i) of the theorem, we deduce that \((\widetilde{X},\widetilde{Y})\) is a Markovian maximal coupling. Furthermore, asis a bijective, bimeasurable function, so application of Lemma 4 to \((t,\widetilde{X}_t) \rightarrow (t,X_t)\) and \((t,\widetilde{Y}_t) \rightarrow (t,Y_t)\) shows that (X, Y) is a Markovian maximal coupling.$$\begin{aligned} (t,x) \mapsto (t, Q^\top (t)(x+l(t)\mathbf {e}_1)) \end{aligned}$$Conversely, let (X, Y) be a Markovian maximal coupling of two copies of the diffusion starting from \(\mathbf {x}_0\) and \(\mathbf {y}_0\). Then the results of Sects. 2.1 and 2.2 show that there exist continuously differentiable functions \(l: [0,\infty ) \rightarrow \mathbb {R}\) and \(\mathbf {n}: [0,\infty ) \rightarrow \mathbb {S}^{d1}\) parametrising the mirror \(I(\mathbf {x}_0,\mathbf {y}_0,t)\). Moreover, these functions should satisfy \(\mathbf {n}(0)=\frac{\mathbf {x}_0\mathbf {y}_0}{\mathbf {x}_0\mathbf {y}_0}\) and \(l(0)=\frac{\mathbf {x}_0^2\mathbf {y}_0^2}{\mathbf {x}_0\mathbf {y}_0}\). To see this, take \(t \downarrow 0\) in (22). Furthermore, Theorem 10 and the corollary following it show that X and Y are coupled on \(t<\tau \) according to the relationshipThe construction of Q follows by applying Gram–Schmidt orthogonalization to extend \(\mathbf {n}(0)\) to an orthonormal basis \((\mathbf {n}(0),\mathbf {v}_1, \ldots , \mathbf {v}_{d1})\) of \(\mathbb {R}^d\). Note that the vectors \(\mathbf {v}_i\) lie in the tangent space of \(\mathbb {S}^{d1}\) based at \(\mathbf {n}(0)\). The vector function \((\mathbf {n}(t):t \ge 0)\) traces out a \(C^1\) curve \(\gamma \) on the sphere \(\mathbb {S}^{d1}\). Parallel transport [14, p. 75] can be applied along \(\gamma \) to each vector \(\mathbf {v}_i\); this produces \(C^1\) vectorfields \(\mathbf {X}_i: [0,\infty ) \rightarrow \mathbb {R}^d\) along \(\gamma \). [14, Proposition 2.74] shows that \((\mathbf {n},\mathbf {X}_1,\ldots ,\mathbf {X}_{d1})\) produces a \(C^1\) orthonormal frame along \(\gamma \), so set$$\begin{aligned} Y_t= ({\mathbb {I}}2\mathbf {n}(t)\mathbf {n}^\top (t))X_t+2l(t)\mathbf {n}(t). \end{aligned}$$(31)We now produce a new pair of diffusions with timevarying drifts, \((\widetilde{X}, \widetilde{Y})\), by applying the transformation (24) to (X, Y) with drift \(\widetilde{\mathbf {b}}\) and driving Brownian motion \(\widetilde{B}\) as described in Lemma 15. This new pair is also a Markovian maximal coupling (use Lemma 4), and from Eq. (31) it follows that the coupled pair \((\widetilde{X},\widetilde{Y})\) is described by the transformation (29). As discussed in part (i) of this proof, the relationship (30) follows as a direct consequence.\(\square \)$$\begin{aligned} Q^\top (t)= (\mathbf {n}(t),\mathbf {X}_1(t),\ldots ,\mathbf {X}_{d1}(t)). \end{aligned}$$
Inverting the relationship (26), and using the relationship (30), the above theorem yields the following characterisation of drifts which permit MMC:
Corollary 18
2.4 Rigidity theorems for timehomogeneous diffusions
The previous subsection established an implicit classification of all timenonhomogeneous diffusions that can be coupled by a Markovian maximal coupling. But, as noted in the literature, not many examples of such couplings are known for timehomogeneous diffusions. It is a matter of general belief that the class of such timehomogeneous diffusions is very small, but little rigorous work appears to have been done to specify this class.
In this subsection we obtain a constraint equation on the drift, leading to certain general conditions on the drift and the starting points which are necessary for the existence of Markovian maximal couplings. In the case of affine drifts the constraint equations are explicit enough to classify all affine drifts leading to Markovian maximal couplings. We then state and prove the main theorem of this subsection: if there are two balls \(\mathcal {B}(\mathbf {x}_0,r)\) and \(\mathcal {B}(\mathbf {y}_0,r)\) in \(\mathbb {R}^d\), such that a Markovian maximal coupling exists from all pairs of points \((\mathbf {x},\mathbf {y}) \in \mathcal {B}(\mathbf {x}_0,r) \times \mathcal {B}(\mathbf {y}_0,r)\), then the drift has to be of a very simple affine form, verifying the popular belief that Markovian maximal couplings are indeed very rare.
We conclude by showing a stronger result for onedimensional diffusions, which states that for such a coupling to exist for a specific pair of starting points, either the drift must be an odd function centred at a point, or it must be affine.
Lemma 19
Proof
Now, suppose \(\mathbf {b}\) satisfies (34) for l and \(\mathbf {n}\) as given in the lemma. Let \(\tau =\inf \{t>0: X_t \in I(\mathbf {x}_0, \mathbf {y}_0,t)\}\). Then (35) shows that \(Y_t=F(t,X_t){\mathbb {I}}(t < \tau ) + X_t {\mathbb {I}}(t \ge \tau )\) gives a valid coupling \(\mu \) of the two copies (X, Y) with coupling time \(\tau \). To see that this is indeed the maximal coupling, obtain the \(C^1\) curve \(Q:[0,\tau ) \rightarrow \mathbf {O}(d)\) from \(\mathbf {n}\) by the procedure given in the proof of Theorem 17 (ii). Now, \((\widetilde{X}, \widetilde{Y})\) obtained from (X, Y) by (24) is reflectioncoupled according to the recipe in (29). Theorem 17 (ii) then implies that (X, Y) is a Markovian maximal coupling. \(\square \)
Lemma 20
 (i)and$$\begin{aligned} S(\mathbf {x})= ({\mathbb {I}}2\mathbf {n}\mathbf {n}^\top )S(F(t,\mathbf {x}))({\mathbb {I}}2\mathbf {n}\mathbf {n}^\top ), \end{aligned}$$(37)In particular, \(S(\mathbf {x})\) and \(S(F(t,\mathbf {x}))\) have the same set of eigenvalues.$$\begin{aligned} T(\mathbf {x})=2(\dot{\mathbf {n}}\mathbf {n}^\top \mathbf {n}\dot{\mathbf {n}}^\top )+({\mathbb {I}}2\mathbf {n}\mathbf {n}^\top )T(F(t,\mathbf {x}))({\mathbb {I}}2\mathbf {n}\mathbf {n}^\top ). \end{aligned}$$(38)
 (ii)There exists a continuous function \(\lambda (\cdot ,\cdot ): [0,\infty ) \times \mathbb {R}^d \rightarrow \mathbb {R}\) such that$$\begin{aligned} \left( \frac{S(\mathbf {x})+S(F(t,\mathbf {x}))}{2}\right) \mathbf {n}=\lambda (t,\mathbf {x})\mathbf {n}. \end{aligned}$$(39)
 (iii)$$\begin{aligned} \left( \frac{T(\mathbf {x})+T(F(t,\mathbf {x}))}{2}\right) \mathbf {n}=\dot{\mathbf {n}}. \end{aligned}$$(40)
Proof
Parts (ii) and (iii) follow by postmultiplying the equations of part (i) by \(\mathbf {n}\), bearing in mind that as \(\mathbf {n}\) is a unit vector therefore \(\mathbf {n}\) and \(\dot{\mathbf {n}}\) must be orthogonal. \(\square \)
Because \({\mathbf {n}(0)=\tfrac{\mathbf {x}_0\mathbf {y}_0}{\mathbf {x}_0\mathbf {y}_0}}\) and \({l(0)=\mathbf {n}(0).\tfrac{\mathbf {x}_0+\mathbf {y}_0}{2}}\), we know \(F(0,\cdot )\) explicitly. Even in the generality of the hypotheses of Lemma 19, one can obtain the following necessary condition on the drift of a Euclidean diffusion for existence of a Markovian maximal coupling: use (ii) of the above lemma and take \(t \downarrow 0\).
Corollary 21
Under the hypotheses of Lemma 19 and (34), \(\mathbf {n}(0)\) must be an eigenvector of \({\tfrac{S(\mathbf {x})+S(F(0,\mathbf {x}))}{2}}\) corresponding to some eigenvalue \(\lambda (\mathbf {x})\), for every \(\mathbf {x}\in \mathbb {R}^d\).
Briefly restrict attention to the case where \(\mathbf {b}(\mathbf {x})\) is affine in \(\mathbf {x}\). The following theorem completely classifies the set of such drifts which ensure Markovian maximal coupling.
Theorem 22
Proof
The finite symmetric matrix \(S\) has discrete spectrum; by this, and the continuity of \(\mathbf {n}(\cdot )\) and \(\lambda (\cdot )\), it follows immediately from (44) that \(\lambda (\cdot ) \equiv \lambda _0\) for some constant \(\lambda _0\). Thus \(\mathbf {n}(t)\), as given by (42), must lie in the eigenspace of S corresponding to \(\lambda _0\), for all time t. Substituting this formula for \(\mathbf {n}(t)\) in Eq. (44) and differentiating (42) k times with respect to t (for \(k=0,1,\ldots , d1\)), then setting \(t=0\), we obtain that the vectors \(T^k(\mathbf {x}_0\mathbf {y}_0)\) for \(0 \le k \le d1\) must all lie in the eigenspace of S corresponding to \(\lambda _0\). As T solves its characteristic equation, it is clear that all the higher powers \(T^k(\mathbf {x}_0\mathbf {y}_0)\) for \(k\ge d\) must also lie in this eigenspace. Using the series representation of \(\mathtt {exp}\left( {Tt}\right) \), this means that \(\mathbf {n}(t)\) must also lie in this eigenspace for all t.
Conversely, suppose there exists an eigenvalue \(\lambda _0\) of S such that the vectors \(T^k(\mathbf {x}_0\mathbf {y}_0)\) (for \(0 \le k \le d1\)) all lie in the eigenspace of S corresponding to \(\lambda _0\). To prove the existence of a Markovian maximal coupling (X, Y) starting from \(\mathbf {x}_0\) and \(\mathbf {y}_0\), we will show that (34) holds with \(\mathbf {n}\) and l as given in the theorem.
The following corollary is immediate from the above theorem.
Corollary 23
If \(d=2\), then under the hypotheses of Theorem 22, A is either a symmetric matrix or of the form \(\lambda _0{\mathbb {I}}+T\) for some real scalar \(\lambda _0\) and a skewsymmetric matrix T.
Proof
If the skewsymmetric part T of A is nonzero, then \(\mathbf {x}_0\mathbf {y}_0\) and \(T(\mathbf {x}_0\mathbf {y}_0)\) are nonzero, mutually orthogonal vectors which lie in the eigenspace of S corresponding to \(\lambda _0\). Thus, this eigenspace is the whole of \(\mathbb {R}^2\) and \(S=\lambda _0{\mathbb {I}}\). \(\square \)
Now, we state and prove the main theorem of this section. Recall the Local Perturbation condition LPCdescribed in the introduction.
Theorem 24
Proof
This idea is made more precise in the following internal lemmas.
Lemma 25
Under the hypotheses of Theorem 24, there exists \(\lambda _0 \in \mathbb {R}\) such that \(S(\mathbf {x})=\lambda _0{\mathbb {I}}\) for all \(\mathbf {x}\in \mathbb {R}^d\).
Proof
Suppse that \(X\) and \(Y\) start at \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0,r)\) and \(\mathbf {y}\in \mathcal {B}(\mathbf {y}_0,r)\) respectively. It follows from letting \(t \downarrow 0\) in part (i) of Lemma 20 that, for all \(\mathbf {z}\in \mathbb {R}^d\), \(S(\mathbf {z})\) and \(S(H(\mathbf {x},\mathbf {y})\mathbf {z})\) have the same set of eigenvalues. (Recall that \(H(\mathbf {x},\mathbf {y})\mathbf {z}\) represents reflection of \(\mathbf {z}\) in the hyperplane \(H(\mathbf {x},\mathbf {y})\).)
Denote \(\mathbf {x}^*=(\mathbf {x}_0+\mathbf {y}_0)/2\) and let \(\mathbf {v}_1=\mathbf {x}_0\mathbf {x}^*\). Extend \(\mathbf {v}_1\) to a basis \(\{\mathbf {v}_1,\ldots , \mathbf {v}_d\}\). If \(\varepsilon \) is sufficiently small then the linearly independent vectors \(\mathbf {n}_i=\mathbf {v}_1+ \varepsilon \mathbf {v}_i, \ i=1,\ldots d\) are such that \(\{\mathbf {x}^*+\mathbf {n}_i: i=1,\ldots d\} \subset \mathcal {B}(\mathbf {x}_0,r)\) and \(\{\mathbf {x}^*\mathbf {n}_i: i=1,\ldots d\} \subset \mathcal {B}(\mathbf {y}_0,r)\). Defining \(\mathbf {x}_i=\mathbf {x}^*+\mathbf {n}_i\) and \(\mathbf {y}_i=\mathbf {x}^*\mathbf {n}_i\), it follows that \(\mathbf {x}^* \in H(\mathbf {x}_i,\mathbf {y}_i)\) for all i. For each \(i\), consider maximally coupled diffusions begun at \((\mathbf {x}_i,\mathbf {y}_i)\): applying part (ii) of Lemma 20 and letting \(t \downarrow 0\), it follows that \(\mathbf {n}_i\) is an eigenvector of \(S(\mathbf {x}^*)\). By construction, no \(\mathbf {n}_i\) is orthogonal to any other \(\mathbf {n}_j\). Since \(S(\mathbf {x}^*)\) is symmetric, it follows that \(\{\mathbf {n}_i: i=1,\ldots , d\}\) correspond to the same eigenvalue, say \(\lambda _0\) and thus, \(S(\mathbf {x}^*)=\lambda _0\mathbb {I}\).
Choosing the set of mirrors \(\mathcal {H}= \mathcal {H}_0\), consider the orbit \(\mathcal {O}(\mathbf {x}^*)\) of \(\mathbf {x}^*\) in \(\mathcal {H}\). If \(\mathcal {O}(\mathbf {x}^*)=\mathbb {R}^d\), then the lemma follows from the previous observation that for any \(\mathbf {z}\in \mathcal {O}(\mathbf {x}^*)\), the set of eigenvalues of \(S(\mathbf {z})\) agrees with that of \(S(\mathbf {x}^*)\).
To see this, let L be the line that passes through \(\mathbf {x}_0\) and \(\mathbf {y}_0\). Let \({\mathbf {v}_0= \tfrac{\mathbf {x}_0 \mathbf {y}_0}{\mathbf {x}_0 \mathbf {y}_0}}\). Write \(\mathbf {x}_{\delta }=\mathbf {x}_0 + \delta \mathbf {v}_0\) and \(\mathbf {y}_{\delta }=\mathbf {y}+ \delta \mathbf {v}_0\) for all \(\delta \in (r,r)\). Thus the mirrors \(h_{\delta }=H(\mathbf {x}_{\delta },\mathbf {y}_{\delta }) \in \mathcal {H}\) for all such \(\delta \), and the orbit of \(\mathbf {x}^*\) under reflection in \(\{h_{\delta }: \delta \in (r,r)\}\) is the whole of L. Thus \(L \subseteq \mathcal {O}(\mathbf {x}^*)\).
Now, for any \(\mathbf {z}\in \mathbb {R}^d\), let H be a plane (dimension of H is two) containing the line L and the point \(\mathbf {z}\). For sufficiently small \(\varepsilon >0\), for all \(\delta \in (\varepsilon ,\varepsilon )\) the mirror \(h_{\delta }'\) containing \(\mathbf {x}^*\) and having normal vector \(\mathbf {v}_{\delta } \in H\) and making an angle \(\delta \) with \(\mathbf {v}_0\) lies in \(\mathcal {H}\). Denote by C the circle centred at \(\mathbf {x}^*\), lying in H and passing through \(\mathbf {z}\). Let \(\mathbf {\hat{z}} \in L \,\cap \, C\). Then the orbit of \(\mathbf {\hat{z}}\) under reflection in \(\{h_{\delta }': \delta \in (\varepsilon ,\varepsilon )\}\) is the whole of C. In particular, \(\mathbf {z}\in \mathcal {O}(\mathbf {x}^*)\). This shows that \(\mathcal {O}(\mathbf {x}^*)=\mathbb {R}^d\) and the lemma follows. \(\square \)
Before proceeding further with the proof of Theorem 24, we record a general fact about real skewsymmetric matrices which follows by spectral decomposition [13].
Lemma 26
If \(\mathcal {N}\) is the null space of a \((d \times d)\) real skewsymmetric matrix T, then \(d\dim (\mathcal {N})\) is even.
We now show that the skewsymmetric part \(T(\mathbf {x})\) is a constant matrix T.
Lemma 27
Under the hypotheses of Theorem 24, \(T(\mathbf {x})\equiv T\) for all \(\mathbf {x}\in \mathbb {R}^d\).
Proof
 Step 1.

If \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0,r)\) and \(\mathbf {y}\in \mathcal {B}(\mathbf {y}_0,r)\), then for all \(\mathbf {z},\mathbf {z}' \in H(\mathbf {x},\mathbf {y})\), \(T(\mathbf {z})=T(\mathbf {z}')\).
Set \({\mathbf {z}^*=\tfrac{\mathbf {z}+ \mathbf {z}'}{2}}\), \({\mathbf {v}_1=\tfrac{\mathbf {z}\mathbf {z}'}{\mathbf {z}\mathbf {z}'}}\) and \({\mathbf {v}_2=\tfrac{\mathbf {x}\mathbf {y}}{\mathbf {x}\mathbf {y}}}\). Extend \(\mathbf {v}_1, \mathbf {v}_2\) to an orthonormal basis \(\mathbf {v}_1,\ldots , \mathbf {v}_d\) of \(\mathbb {R}^d\). Using the method of the proof of Lemma 25, construct independent vectors \(\mathbf {n}_i=\mathbf {v}_2 + \varepsilon \mathbf {v}_i, \ i=2,\ldots d\), choosing \(\varepsilon >0\) small enough so thatfor all \(i=2,\ldots ,d\). Writing \(\mathbf {x}_i=\mathbf {z}^*+\mathbf {n}_i\) and \(\mathbf {y}_i=\mathbf {z}^*\mathbf {n}_i\), and with a possibly smaller choice of \(\varepsilon >0\), the hyperplane \(H(\mathbf {x}_i,\mathbf {y}_i)\) lies in \(\mathcal {H}_0\) and the line joining \(\mathbf {z}\) and \(\mathbf {z}'\) is contained in \(H(\mathbf {x}_i,\mathbf {y}_i)\) for all \(i=2,\ldots ,d\). Thus, \(H(\mathbf {x}_i,\mathbf {y}_i)\mathbf {z}=\mathbf {z}\) and \(H(\mathbf {x}_i,\mathbf {y}_i)\mathbf {z}'=\mathbf {z}'\) for all \(i=2,\ldots ,d\). Taking \(t \downarrow 0\) in part (iii) of Lemma 20, it follows that$$\begin{aligned} H(\mathbf {z}^*+\mathbf {n}_i, \mathbf {z}^*\mathbf {n}_i)\mathbf {x}\in \mathcal {B}(\mathbf {y}_0,r) \end{aligned}$$for all \(i=2,\ldots ,d\), implying \(d\mathcal {N}(T(\mathbf {z})T(\mathbf {z}')) \le 1\). Together with Lemma 26, this establishes Step 1.$$\begin{aligned} (T(\mathbf {z})T(\mathbf {z}'))\mathbf {n}_i=0 \end{aligned}$$  Step 2.

There is \(\varepsilon >0\) such that \(T(\mathbf {z})=T(\mathbf {z}')\) for all \(\mathbf {z},\mathbf {z}' \in \{\mathbf {w} \in \mathbb {R}^d\;:\; {\text {dist}}(\mathbf {w}, H(\mathbf {x}_0,\mathbf {y}_0)) < \varepsilon \}\), where \({\text {dist}}(\mathbf {w},A)\) denotes the distance of \(\mathbf {w}\) from the set A.
Choose \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0,r)\) such that the vector \(\mathbf {x}\mathbf {y}_0\) is not parallel to \(\mathbf {x}_0\mathbf {y}_0\). It follows from Step 1 that \(T(\mathbf {z})=T(\mathbf {z}')\) for all \(\mathbf {z},\mathbf {z}' \in H(\mathbf {x},\mathbf {y})\). Choose \(\varepsilon >0\) such that \({\mathbf {y}_{\delta }=\mathbf {y}_0 +\delta \tfrac{\mathbf {x}_0 \mathbf {y}_0}{\mathbf {x}_0 \mathbf {y}_0} \in \mathcal {B}(\mathbf {y}_0,r)}\) for all \(\delta \in (2 \varepsilon , 2\varepsilon )\). Note that the vector \(\mathbf {x}\mathbf {y}_\delta \) is not parallel to \(\mathbf {x}_0\mathbf {y}_0\) for any \(\delta \in (2 \varepsilon , 2\varepsilon )\). Using Step 1 again, \(T(\mathbf {z})=T(\mathbf {z}')\) for all \(\mathbf {z},\mathbf {z}' \in H(\mathbf {x}_0,\mathbf {y}_{\delta })\). The assertion now follows from Step 1 and the fact that \(H(\mathbf {x}_0,\mathbf {y}_{\delta }) \,\cap \, H(\mathbf {x},\mathbf {y})\) is nonempty for each \(\delta \in (2 \varepsilon , 2\varepsilon )\).
 Step 3.
 Now we work with the set of mirrorswhere \(\varepsilon \) is chosen as in Step 2. For notational convenience, we write \(h_{\delta }=H(\mathbf {x}_0,\mathbf {y}_{\delta })\). The \(\mathbf {y}_\delta =\mathbf {y}_0 +\delta \tfrac{\mathbf {x}_0 \mathbf {y}_0}{\mathbf {x}_0 \mathbf {y}_0}\) all lie on the same line through \(\mathbf {x}_0\), and therefore all these mirrors have a common normal vector, which we write \(\mathbf {n}^*\). Let \((l_{\delta },\mathbf {n}_{\delta })\) parametrize the interface \(I(\mathbf {x}_0, \mathbf {y}_{\delta }, \cdot )\) corresponding to the starting points \(\mathbf {x}_0\) and \(\mathbf {y}_{\delta }\) of the diffusions X and Y respectively. For each \(\delta \), \(\mathbf {n}_{\delta }(0)=\mathbf {n}^*\). Furthermore, by letting \(t \downarrow 0\) in part (iii) of Lemma 20,$$\begin{aligned} \mathcal {H}=\left\{ H(\mathbf {x}_0,\mathbf {y}_{\delta }): \delta \in (2\varepsilon , 2\varepsilon )\right\} , \end{aligned}$$Given \(\delta \in (2\varepsilon , 2\varepsilon )\), the distance of the point \({\tfrac{\mathbf {x}_0+\mathbf {y}_{\delta }}{2}}\) from the hyperplane \(H(\mathbf {x}_0, \mathbf {y}_0)\) is less than \(\varepsilon \). Consequently Step 2 implies that \(\dot{\mathbf {n}}_{\delta }(0)=\dot{\mathbf {n}}_0(0)=\mathbf {n}'\) (say) for all \(\delta \in (2\varepsilon , 2\varepsilon )\).$$\begin{aligned} \dot{\mathbf {n}}_{\delta }(0)= T\left( \tfrac{\mathbf {x}_0+\mathbf {y}_{\delta }}{2}\right) \mathbf {n}^*. \end{aligned}$$
Choose any \(\mathbf {z}, \mathbf {z}' \in \mathbb {R}^d\) such that \({\mathbf {z}'=\mathbf {z}+\delta \tfrac{\mathbf {x}_0 \mathbf {y}_0}{\mathbf {x}_0 \mathbf {y}_0}}\) for some \(\delta \in (2 \varepsilon , 2\varepsilon )\). Set \(\mathbf {z}^*=h_0\mathbf {z}\) so that \(\mathbf {z}=h_0\mathbf {z}^*\). Noting that \(\mathbf {z}\), \(\mathbf {z}^*\), \(\mathbf {z}'\) lie on the same line perpendicular to \(H(\mathbf {x}_0, \mathbf {y}_0)\), it follows from an argument about onedimensional reflections that \(\mathbf {z}'=h_{\delta }\mathbf {z}^*\).
Then, by part (i) of Lemma 20, we getfrom which we get$$\begin{aligned} T(\mathbf {z}^*)= & {} 2(\mathbf {n}'\mathbf {n}^{*\top }\mathbf {n}^*\mathbf {n}'^\top )+({\mathbb {I}}2\mathbf {n}^*\mathbf {n}^{*\top })T(\mathbf {z})({\mathbb {I}}2\mathbf {n}^*\mathbf {n}^{*\top })\nonumber \\= & {} 2(\mathbf {n}'\mathbf {n}^{*\top }\mathbf {n}^*\mathbf {n}'^\top )+({\mathbb {I}}2\mathbf {n}^*\mathbf {n}^{*\top })T(\mathbf {z}')({\mathbb {I}}2\mathbf {n}^*\mathbf {n}^{*\top }) \end{aligned}$$(48)which gives \(T(\mathbf {z})=T(\mathbf {z}')\). Hence the lemma follows. \(\square \)$$\begin{aligned} ({\mathbb {I}}2\mathbf {n}^*\mathbf {n}^{*\top })(T(\mathbf {z})T(\mathbf {z}'))({\mathbb {I}}2\mathbf {n}^*\mathbf {n}^{*\top })=0 \end{aligned}$$
Lemmas 25 and 27 together are sufficient to prove Theorem 24. \(\square \)
Theorem 24 can be strengthened if \(\dot{\mathbf {n}}(t)=0\) for all t, i.e., the interface translates but does not rotate in time. We state this in the following theorem. Since there is no rotation, the driving Brownian motions in the stochastic differential equation for X and Y are constant reflections of each other. So we can assume without loss of generality that \(l(0)=0\) and \(\mathbf {n}(t) \equiv \mathbf {e}_1\).
Theorem 28
 (i)\(l(t) =0\) for all \(t\ge 0\), in which case the drift vectorfield \(\mathbf {b}\) must satisfyfor all \(\mathbf {x}\in \mathbb {R}^d\).$$\begin{aligned} \mathbf {b}(h_1\mathbf {x})=h_1\mathbf {b}(\mathbf {x}) \end{aligned}$$
 (ii)\(l(t) \ne 0\) for some \(t>0\), in which case the drift vectorfield \(\mathbf {b}\) must satisfyfor all \(\mathbf {x}=(x_1,\mathbf {x}^{(1)}) \in \mathbb {R}^d\), where \(c_1, c_2\) are constants and \(\mathbf {f}: \mathbb {R}^{d1} \rightarrow \mathbb {R}^{d1}\) is continuously differentiable.$$\begin{aligned} \mathbf {b}(x_1, \mathbf {x}^{(1)})=\left( c_1x_1 + c_2, \mathbf {f}(\mathbf {x}^{(1)})\right) ^\top \end{aligned}$$
Proof
Part (i) follows from the fact that the generators of Y and \(h_1X\) are the same.
The case of onedimensional diffusions is a trivial consequence of the above theorem, as noted in the next corollary.
Corollary 29
Assume (A1) and (A2) hold for a onedimensional timehomogeneous Euclidean diffusion. Then there exists a Markovian maximal coupling of X and Y starting from \(x_0\) and \(y_0\) respectively if and only if either the drift vectorfield b is affine or it obeys the reflection symmetry \(b(x)=b(x_0+y_0x)\) for all \(x \in \mathbb {R}\).
Remark 30
3 Markovian maximal couplings for manifolds
In this section, we analyse rigidity phenomena for Markovian maximal couplings (MMC) for smooth elliptic diffusions, and demonstrate that there are powerful geometric consequences arising from a natural connection to the theory of diffusion processes on manifolds (specifically, the notion of Riemannian Brownian motion with drift). The main task of this section is to understand how the Euclidean arguments of Sect. 2 carry over to the manifold case. In particular, the existence of Markovian maximal couplings (together with LPC) has profound rigidity consequences for the geometry of the manifold.
 (i)
The Riemannian manifold (M, g) obtained above is complete (we say that the diffusion \(X\) is diffusiongeodesic complete). This is a purely technical assumption and the completeness is usually not too hard to check as we know the diffusion coefficients explicitly. In particular, diffusiongeodesic completeness trivially holds on compact manifolds. Diffusiongeodesic completeness is not a necessary condition for the existence of Markovian maximal couplings, as can be seen for dimension \(d \ge 2\) by considering reflection couplings of Brownian motions on the ddimensional punctured sphere \(\mathbb {S}^d  \{P\}\) obtained by deleting a point P from the sphere \(\mathbb {S}^d\) (and the corresponding couplings of diffusions obtained on the plane by stereographic projection). In this example, the existence of a rich supply of MMC follows from the fact that this space has a completion \(\mathbb {S}^d\) on which we can construct MMC of Brownian motions started from any two points (see [25]), and from the fact that if \(d \ge 2\) then the Brownian motion started in \(\mathbb {S}^d  \{P\}\) almost surely does not hit P. It is an interesting question whether this is the ‘generic’ example for instances where diffusiongeodesic completeness fails but Markovian maximal couplings exist, raising issues which seem somewhat reminiscent of the topic of resolution of singularities in algebraic geometry. We hope to address this in a future article.
 (ii)
Our diffusion process X is defined for all time. This is to ensure that we are dealing with probability densities which is essential for the arguments in Sect. 1.1 to go through. For Brownian motion on M, this can be resolved by ensuring that M is stochastically complete. There are a number of intrinsic geometric properties of M that ensure stochastic completeness, such as the existence of a constant lower bound on the Ricci curvature. See [17], for example, for more details.
3.1 Brownian motion with drift on the manifold
Not only can any smooth elliptic diffusion on M be written as Brownian motion with drift on (M, g), but also this permits a rather explicit geometric construction of the diffusion which facilitates the discussion of probabilistic coupling techniques, namely the Eells–Elworthy–Malliavin construction [12].
This framework provides an expressive way to define smooth elliptic diffusions (and other semimartingale processes) on M, as follows.
Note that, when \(\mathbf {b}=0\), the above construction reduces to the classical EellsElworthyMalliavin construction of Brownian motion on M.
3.2 Couplings of diffusions on manifolds
Once we have the above construction, a natural question to ask is: when is there a Markovian maximal coupling (MMC) for two copies of the diffusion starting from \(\mathbf {x}_0\) and \(\mathbf {y}_0\)? In the Euclidean case there is a complete characterization of the class of timehomogeneous diffusions under LPC, which is to say, when two copies of the diffusion can be maximally coupled whenever they start from \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0, r)\) and \(\mathbf {y}\in \mathcal {B}(\mathbf {y}_0, r)\) (for \(\mathcal {B}(\mathbf {x}_0, r)\) and \(\mathcal {B}(\mathbf {x}_0, r)\) chosen to be two arbitrary disjoint open balls in \(\mathbb {R}^d\)). Theorem 24 shows that the class of such diffusions is actually very small.
The proof of Theorem 24 depends strongly on a wealth of isometries of Euclidean space arising via iterated reflections. Very few other ddimensional Riemannian manifolds have many isometries, and so we may expect an even stronger rigidity phenomenon to hold for the geometry of (nonEuclidean) manifolds on which there is a good supply of MMC. The work of this section substantiates this expectation.
We begin by recalling briefly some notions from the Euclidean case (Sect. 2). We have noted that the Local Perturbation Condition LPC (Definition 1) makes sense for any metric space, including the Riemannian manifold case. Let X and Y be two copies of the elliptic diffusion derived from the stochastic differential equation (56), and starting from \(\mathbf {x}_0\) and \(\mathbf {y}_0\) respectively. Note that the assumptions of ellipticity and smoothness of the coefficients of L together ensure that the law of X (equivalently Y) has a smooth positive density with respect to the Riemannian volume measure m for every positive time \(t>0\), which we write as \(p(\mathbf {x}_0;t,\mathbf {z})\), \(p(\mathbf {y}_0;t,\mathbf {z})\) for \(t > 0\), \(\mathbf {z}\in M\).
We suppose that the standing assumptions of diffusiongeodesic completeness and stochastic completeness both hold for the regular elliptic diffusion \(X\), so that the resulting Riemannian manifold \(M\) is geodesically complete and so that \(X\) stays on \(M\) for all time. Thus from here on we are considering the case of Brownian motion with nonexplosive drift on a complete Riemannian manifold.
We note here that all the results in Sect. 1.1 carry over to the manifold setting with \((M,{\text {dist}})\) being the Riemannian manifold (with the distance \({\text {dist}}\) induced by the Riemannian metric) and m taken to be the volume measure.
3.3 The interface
Varadhan smalltime asymptotics and Lemma 3 can be used to show the following: that the existence of an MMC implies that, for each time t, there is a deterministic involutive isometry \(F_t\) which exchanges \(X_t\) with \(Y_t\) and fixes the set of points equidistant from both \(X_t\) and \(Y_t\). This generalizes the timevarying reflection isometry of Euclidean space which is mentioned in Remark 11; the fixedpoint set of \(F_t\) corresponds to the ‘evolving mirror’ of the Euclidean case.
The rôle of Varadhan’s smalltime asymptotics in the following is analogous to the rôle of Lemma 7 in the Euclidean case. This powerful technique gives the logarithmic asymptotics of the density of \(X_t\) when \(t \downarrow 0\), as stated in the following lemma.
Lemma 31
This theorem was proven by [39] for diffusion processes on Euclidean space. Later [29] noticed that Varadhan’s arguments carry over to diffusions on closed manifolds whose generators are of the form \(L=\frac{1}{2}\Delta _M + \mathbf {b}\). Molchanov also showed that this result could be extended to general smooth complete manifolds by introducing a reflected diffusion in a suitably large domain \(U \subset M\) containing \(\mathbf {x}\) and \(\mathbf {y}\), with the same generator \(L\) inside, and using this process to define a natural diffusion on the ‘double’ U. He then showed that smoothing techniques allowed the approximation of the ‘double’ U by a smooth closed manifold, such that the diffusion thus defined has a density that is sufficiently close to that of the original one [29, p. 18 and further references].
We can now restate the pivotal Theorem 10 from Sect. 2.1 in the new context of manifolds. The proof of the manifold case follows that of the Euclidean case, but uses Lemma 31 in place of Lemma 7, and uses the strong maximum principle (Lemma 9) in local coordinates; we omit details.
Theorem 32
Let \(\tau '=\inf \{s>0: X_s \in I(\mathbf {x}_0,\mathbf {y}_0,s)\}\) be the first time that \(X\) hits the interface. Then the following holds.
Corollary 33
Proof
The proof follows the lines of the proof of Corollary 12. The only additional detail that we have to check here (which was immediate in the Euclidean case) is that, for any \(t > 0\) with \(X_t \ne Y_t\), any \(\mathbf {z}\in H(X_t,Y_t)\) and any rational sequence \(t_n \downarrow t\), there is \(\mathbf {z}_n \in H(X_{t_n},Y_{t_n})\) such that \(\mathbf {z}_n \rightarrow \mathbf {z}\). This was used in Corollary 12 to show \(H(X_t,Y_t) \subseteq I(\mathbf {x}_0,\mathbf {y}_0,t))\).
Recall the event \(E=\cap _{q \in Q}E_q\), where \(E_q\) was defined in (19). Assume E holds. For notational convenience, denote \(H(X_t,Y_t), X_t, Y_t\) by \(H,\mathbf {x},\mathbf {y}\) and \(H(X_{t_n},Y_{t_n}), X_{t_n}, Y_{t_n}\) by \(H_n,\mathbf {x}_n,\mathbf {y}_n\) respectively. Let \(\gamma :[0,2{\text {dist}}(\mathbf {x},\mathbf {z})] \rightarrow M\) denote the continuous curve such that \(\gamma \mid _{[0,{\text {dist}}(\mathbf {x},\mathbf {z})]}\) is a minimal geodesic joining \(\mathbf {x}\) and \(\mathbf {z}\) and \(\gamma \mid _{[{\text {dist}}(\mathbf {x},\mathbf {z}),2{\text {dist}}(\mathbf {x},\mathbf {z})]}\) is a minimal geodesic joining \(\mathbf {z}\) and \(\mathbf {y}\). As M has no branching geodesics, it follows that \({\text {dist}}(\mathbf {x},\gamma (s)) < {\text {dist}}(\mathbf {y},\gamma (s))\) for any \(s \in [0,{\text {dist}}(\mathbf {x},\mathbf {z}))\). Consequently for any \(\delta >0\), by the compactness of \(\{\gamma (s) : s \in [0,{\text {dist}}(\mathbf {x},\mathbf {z})\delta ]\}\), \(\min _{s \in [0,{\text {dist}}(\mathbf {x},\mathbf {z})\delta ]}({\text {dist}}(\mathbf {y}, \gamma (s)){\text {dist}}(\mathbf {x}, \gamma (s))) > 0\) and hence, \(\min _{s \in [0,{\text {dist}}(\mathbf {x},\mathbf {z})\delta ]}({\text {dist}}(\mathbf {y}_n, \gamma (s)){\text {dist}}(\mathbf {x}_n, \gamma (s))) > 0\) for sufficiently large n. Thus, for sufficiently large n, \(\gamma (s) \in H^(\mathbf {x}_n,\mathbf {y}_n)=I^(\mathbf {x}_0,\mathbf {y}_0,t_n)\) for all \(s \in [0,{\text {dist}}(\mathbf {x},\mathbf {z})\delta ]\) and consequently, \(\min _{s \in [0,{\text {dist}}(\mathbf {x},\mathbf {z})\delta ]}\alpha (t_n,\gamma (s))>0\). Similarly, \(\min _{s \in [{\text {dist}}(\mathbf {x},\mathbf {z})+\delta ,2{\text {dist}}(\mathbf {x},\mathbf {z})]}\alpha (t_n,\gamma (s))<0\) for sufficiently large n. Thus, as E holds, the continuity of \(\alpha (t_n, \cdot )\), implies that for sufficiently large n, there is \(\mathbf {z}_n \in \gamma \,\cap \, H_n\) such that \(\mathbf {z}_n \rightarrow \mathbf {z}\). As \(\mu (E)=1\), this implies \(H(X_t,Y_t) \subseteq I(\mathbf {x}_0,\mathbf {y}_0,t))\) almost surely.
The rest of the proof carries over verbatim from that of Corollary 12. \(\square \)
The striking fact that emerges from the above is that, almost surely under the coupling \(\mu \), for each \(s>0\), \(H(X_t,Y_t)\) is a nonrandom set which depends only on s and not on the specific location of \((X_t,Y_t)\). We will call this set \(H_t\) henceforth. Similarly, denote \(H^+_t=H^+(X_t,Y_t)\) and \(H^_t=H^(X_t, Y_t)\). The family \(\{H_t: t \ge 0\}\) corresponds to the family of moving mirrors from Sect. 2.
We now follow [25]’s construction to define a deterministic global involutive isometry \(F_s\) which fixes \(H_s\) and maps \(X_s\) to \(Y_s\) under the coupling. The argument of [25, Lemma 4.6] applies directly to our case: we therefore omit proof.
Lemma 34
Whenever such a \(\mathbf {y}\) exists, we will call \(\mathbf {y}\) the mirror image of \(\mathbf {x}\) at time s. With the aid of the above lemma, the isometry \(F_s\) is constructed using a procedure which is similar to [25, Theorem 4.5], but is subject to some modification as described in the following lemma and its proof.
Lemma 35
Suppose that the standing assumptions of diffusiongeodesic completeness and stochastic completeness both hold. Assume (X, Y) is a Markovian maximal coupling with starting points \(\mathbf {x}_0\) and \(\mathbf {y}_0\). Then, for each \(s \in [0,\tau )\), there is a deterministic involutive isometry \(F_s\) with fixed point set \(H_s\) such that \(Y_s=F_s(X_s)\), furthermore \(F_s(H^_s)=H^+_s\).
Proof
Thus \(F_s\) is defined on the whole of M for every \(s \ge 0\). Continuity of \(F_s\) for \(s \ge 0\) follows exactly along the lines of the proof of continuity of the map R in [25, Theorem 4.5]. Further, by definition, \(F_s\) is involutive. Thus, in particular, \(F_s\) is an open map.
To prove that \(F_s\) is, in fact, an isometry, we have to modify the proof of [25, Lemma 5.3] appropriately, as we outline in the following.
Now, \(F_s(H^_s)=H^+_s\) follows from Lemma 34. This completes the proof of the lemma. \(\square \)
3.4 Structure of the manifold M
In this section, we will use the isometries \(f_{\mathbf {x},\mathbf {y}}\) constructed above for every pair of points \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0,r)\) and \(\mathbf {y}\in \mathcal {B}(\mathbf {y}_0, r)\) to show that the underlying complete Riemannian manifold M is homogeneous (i.e. the isometry group acts transitively) and isotropic about a chosen point \(\mathbf {x}^*\) (i.e. there are \(\tfrac{d(d1)}{2}\) independent rotations about \(\mathbf {x}^*\)). This will imply that M is a maximally symmetric space, i.e. the isometry group \(\mathcal {G}\) of M has the maximal dimension possible (namely, \(\tfrac{d(d+1)}{2}\)) for any ddimensional manifold. It is an almost immediate consequence that the space M can be classified (up to scaling) as one of the three model space forms of constant curvatures respectively \(1\), \(0\), and \(+1\).
Lemma 36
Suppose that the standing assumptions of diffusiongeodesic completeness and stochastic completeness both hold. Under LPC, (M, g) is a homogeneous space.
Proof
In the following lemma, we will write \(\mathbf {x}^*\) for the midpoint of a minimal geodesic \(\gamma _{\mathbf {x}_0,\mathbf {y}_0}\) connecting \(\mathbf {x}_0\) and \(\mathbf {y}_0\). If two vectors u, v belong to the same tangent space then we denote the angle between them by \(\angle (u,v)\).
Lemma 37
Suppose that the standing assumptions of diffusiongeodesic completeness and stochastic completeness both hold. Under LPC, M is isotropic at \(\mathbf {x}^*\).
Proof
Let \(\gamma (v)\) denote the geodesic issuing from \(\mathbf {x}^*\) in direction v. Suppose \(\gamma (v_0)=\gamma _{\mathbf {x}_0,\mathbf {y}_0}\), thus defining a unit vector \(v_0\). The proof proceeds in three steps as follows.
Step 1. First, we want to show that there is \(\varepsilon >0\) such that, for any \(v \in T_{\mathbf {x}^*}M\) with \(\angle (v,v_0) < \varepsilon \), there is an isometry \(g_v\) leaving \(\mathbf {x}^*\) fixed and \({\text {d}}g_v(v_0)=v\).
By continuity of geodesics in the starting direction, we can choose \(\varepsilon >0\) sufficiently small so that \(\gamma (v')\) intersects \(\mathcal {B}(\mathbf {x}_0, r)\) and \(\gamma (v')\) intersects \(\mathcal {B}(\mathbf {y}_0, r)\) whenever \(\angle (v',v_0) < \varepsilon \). By [32, Proposition 20, p. 141], with a possibly smaller choice of \(\varepsilon >0\), we can take \(\mathbf {x}_{v'} \in \gamma (v') \,\cap \, \mathcal {B}(\mathbf {x}_0,r)\) and \(\mathbf {y}_{v'} \in \gamma (v') \,\cap \, \mathcal {B}(\mathbf {y}_0,r)\) such that \(\gamma (v')\) realises the distance \({\text {dist}}(\mathbf {x}^*,\mathbf {x}_{v'})\) and \(\gamma (v')\) realises the distance \({\text {dist}}(\mathbf {x}^*,\mathbf {y}_{v'})\). Furthermore, by continuity of the metric, when \(\varepsilon >0\) is small enough, we can take such \(\mathbf {x}_{v'}\), \(\mathbf {y}_{v'}\) satisfying \({\text {dist}}(\mathbf {x}_{v'},\mathbf {x}^*)={\text {dist}}(\mathbf {y}_{v'},\mathbf {x}^*)\) whenever \(\angle (v',v_0) < \varepsilon \). Thus, from the developments of the previous subsection, there is an involutive isometry \(f_{\mathbf {x}_{v'},\mathbf {y}_{v'}}\) which fixes \(\mathbf {x}^*\), inverts the geodesic passing through \(\mathbf {x}^*\) in direction \(v'\), and fixes all the geodesics which pass through \(\mathbf {x}^*\) in directions orthogonal to \(v'\).
Now, take any unit vector \(v \in T_{\mathbf {x}^*}M\) with \(\angle (v,v_0)<2\varepsilon \). Let \({v'=\tfrac{v+v_0}{v+v_0}}\). By the properties of rhombuses, \(\angle (v',v_0)=\tfrac{1}{2}\angle (v,v_0) < \varepsilon \), and thus \(f_{\mathbf {x}_{v'},\mathbf {y}_{v'}}\) exists as specified in the preceding paragraph. Now, consider the isometry \(g_v= f_{\mathbf {x}_{v'},\mathbf {y}_{v'}}\circ f_{\mathbf {x}_0,\mathbf {y}_0}\). Note that \(g_v\) fixes \(\mathbf {x}^*\) and a straightforward calculation reveals \({\text {d}}g_v(v_0)=v\). This \(g_v\) is our required isometry.
Step 2. Take any unit vector \(w \in T_{\mathbf {x}^*}M\) such that w and \(v_0\) are linearly independent. Let \(\Pi \) be the twodimensional subspace of \(T_{\mathbf {x}^*}M\) generated by \(v_0\) and w and denote by \(\mathbb {S}(v_0,w)\) the circle in \(T_{\mathbf {x}^*}M\) centred at the origin of \(T_{\mathbf {x}^*}M\) and running through \(v_0\) and w. Let U be a normal neighbourhood around \(\mathbf {x}^*\). Let \(S_{\Pi }=\exp _{\mathbf {x}^*}(\Pi ) \,\cap \, U\) denote the twodimensional fragment of M corresponding to \(\Pi \) and lying in \(U\).
Denote by \(\mathcal {H}(v_0,w)\) the closed subgroup of isometries generated by \(\{g_v: v \in \mathbb {S}(v_0, w)\),
\(\angle (v,v_0) < \varepsilon \},\) where \(g_v\) are the isometries constructed in Step 1. Note that the set \(\{g_v: v \in \mathbb {S}(v_0, w)\),
Note that, if \(v_n= {\text {d}}g_n(v_0)\) such that \(v_n \rightarrow v\), then, by the fact that \(g_n(\mathbf {x}^*)=\mathbf {x}^*\) for all n, we can choose a subsequence \(g_{n_k}\) and a \(g \in \mathcal {H}(v_0,w)\) such that \(g_{n_k} \rightarrow g\) in the topology of isometries [30, p. 7]. Thus, by [30, Lemma 4], \(dg_{n_k}(v_0) \rightarrow dg(v_0)\) implying \(O(v_0)\) is closed. Furthermore, if \(g \in \mathcal {H}(v_0,w)\) then dg is a linear isometry on \(T_{\mathbf {x}^*}M\). So the same argument as in the previous lemma shows that \(O(v_0)\) is open. Thus, \(O(v_0)=\mathbb {S}(v_0, w)\).
Thus, in particular, the subgroup of isometries \(\mathcal {G}_{\mathbf {x}^*}\) which fix \(\mathbf {x}^*\) (the isotropy group at \(\mathbf {x}^*\)) generates all the rotations of \(T_{\mathbf {x}^*}M\) based at \(\mathbf {x}^*\) in 2planes containing \(v_0\). We describe the isometries in \(\mathcal {H}(v_0,w)\) as rotations in \(\mathbb {S}(v_0, w)\).
Step 3. We will now show that, given two ordered orthonormal frames based at \(T_{\mathbf {x}^*}M\), there is a sequence of isometries in \(\mathcal {G}_{\mathbf {x}^*}\) that take one to the other. In particular this implies that M is isotropic at \(\mathbf {x}^*\). Let \((e_1, \ldots , e_d)\) and \((e'_1, \ldots , e'_d)\) be ordered orthonormal frames in \(T_{\mathbf {x}^*}M\). We can apply rotations in \(\mathbb {S}(v_0, e_1)\) (respectively \(\mathbb {S}(v_0, e'_d)\)) to align \(e_1\) with \(v_0\) (respectively \(e'_d\) with \(v_0\)). Thus, without loss of generality, we consider frames of the form \((v_0, e_2, \ldots , e_d)\) and \((e'_1, \ldots , e'_{d1},v_0)\).
Now, apply a rotation in \(\mathbb {S}(v_0,e'_1)\) to transform \((v_0, e_2, \ldots , e_d)\) to \((e'_1,e^{(1)}_2 \ldots , e^{(1)}_d)\) for some unit vectors \(e^{(1)}_2, \ldots , e^{(1)}_d\) in \(T_{\mathbf {x}^*}M\). If \(v_0\) and \(e^{(1)}_2\) are linearly independent, then apply a rotation in \(\mathbb {S}(v_0,e^{(1)}_2)\), to bring \((e'_1,e^{(1)}_2, \ldots , e^{(1)}_d)\) to \((e'_1,v_0, e^{(2)}_3, \ldots , e^{(2)}_d)\). If \(e^{(1)}_2=v_0\), then achieve the same result using the reflection \(f_{\mathbf {x}_0,\mathbf {y}_0}\). Note that these operations both keep \(e'_1\) fixed as it is orthogonal to \(\{v_0, e^{(1)}_2\}\).
The same procedure is applied inductively to \((e'_1,v_0, e^{(2)}_3, \ldots , e^{(2)}_d)\) to obtain \((e'_1, e'_2, v_0, e^{(4)}_4, \ldots , e^{(4)}_d)\) (note that these operations leave \(e'_1\) fixed), and so on. Finally we obtain \((e'_1, \ldots , e'_{d1}, v_0)\), which proves the lemma. \(\square \)
The above two lemmas imply the following rigidity theorem which completely classifies the space M.
Theorem 38
Suppose that the complete, connected Riemannian manifold \(M\) supports Brownian motion with drift for which there is a Markovian maximal coupling and moreover LPC holds. Then M has constant sectional curvature. Moreover M must be simply connected and therefore (up to scaling) M must be one of the three model spaces \(\mathbb {R}^d\), \(\mathbb {S}^d\) and \(\mathbb {H}^d\).
Proof
By Lemmas 36 and 37, we see that M is a maximally symmetric space, i.e., the dimension of \(\text {Iso}(M)\) is \(\frac{d(d+1)}{2}\) [36, p. 195]. In particular, this implies that \(M\) has constant sectional curvature [32, p. 190]. For the second part of the corollary, the argument of [32, p. 190] shows that a complete, connected maximally symmetric Riemannian manifold must be one of the three model spaces above, or \(\mathbb {RP}^d\). But, as observed in [25, Example 6.4], there is no involutive isometry of \(\mathbb {RP}^d\) of the form described in Lemma 35. This proves the theorem. \(\square \)
Remark 39
For the three model spaces described above, for every \(\mathbf {x}, \mathbf {y}\in M\), the reflection isometry \(f_{\mathbf {x},\mathbf {y}}\), and hence the set of its fixed points \(H(\mathbf {x},\mathbf {y})\), can be explicitly described (see, for example, [24, Example 4.6]). It follows from this explicit description that the submanifold \(H(\mathbf {x},\mathbf {y})\) with the induced metric is again one of the three model spaces with the same curvature as the ambient manifold M and having codimension one.
3.5 Evolution of the mirror isometries
Having classified the space \(M\), we must now classify the set of drift vectorfields \(\mathbf {b}\) which permit MMC with LPC. This necessitates analysis of the evolution of the isometries \(F_s\) as s varies. As noted above, [30] proved that the set of isometries \(\mathcal {G}\) has the structure of a Lie group. The first objective is to prove that the curve of isometries \((F_s:s \ge 0)\) is a \(C^1\) curve in this Lie group.
Lemma 40
Suppose that the standing assumptions of diffusiongeodesic completeness and stochastic completeness both hold. The curve \(s \mapsto F_s\) is a \(C^1\) curve in the Lie group \(\mathcal {G}\).
Proof
Recall that any point in M has a neighbourhood, called a \(\sigma \)neighbourhood, such that any point in this neighbourhood is in a normal coordinate ball of any other point in the same neighbourhood. We study continuity and continuous differentiability of \((F_s:s \ge 0)\) at \(s=t\). As we are investigating a local property, we work in two separate sets of normal coordinates; one set describing a \(\sigma \)neighbourhood U around \(\mathbf {x}\) and the other set describing another \(\sigma \)neighbourhood V around \(F_t(\mathbf {x})\) such that \(F_t(\overline{U}) \subset V\).
The first step is to prove that \(s \mapsto F_s\) is continuous in \(\mathcal {G}\) at \(s=t<\tau \). To show this, it suffices to show that any set of \(d+1\) points \(\mathbf {x}_i \in M\), all of which lie in a \(\sigma \)neighbourhood and are linearly independent (i.e. do not belong in the same \((d1)\)dimensional geodesic hypersurface), produces continuous curves \(s \mapsto F_s(\mathbf {x}_i)\) in M [30]. We note here that we can obtain such a set of \(d+1\) points in any dense subset of any open set in M. To show the continuity of these curves, we will use the continuity of the diffusion paths and the fact that, by Corollary 33, \(Y_s=F_s(X_s)\) when \(s < \tau \).
It is necessary to address the question of rightcontinuity at \(t=0\). Take \(\mathbf {x}\in H^_0\) and consider the case when \(t_n \downarrow 0\). Take a sequence \(\mathbf {x}_n \rightarrow \mathbf {x}\) such that \(\mathbf {x}_n \in H^_{t_n}\). An argument following the treatment of the case \(s=0\) in the proof of Lemma 35 shows that \(F_{t_n}(\mathbf {x}_n) \rightarrow F_0(\mathbf {x})\). As \(F_{t_n}\) is an isometry for each n, we can deduce that \(F_{t_n}(\mathbf {x}) \rightarrow F_0(\mathbf {x})\), thus proving rightcontinuity.
The next step is to prove differentiability at \(t>0\). With \(\sigma \)neighbourhoods U, V of \(\mathbf {x}\), \(F_t(\mathbf {x})\) as described above, let \(\tau _U=\inf \{s \ge t: X_s \notin U\}\). Because the coupling is Markovian, \(\tau _U\) is a stopping time with respect to the filtration generated by the coupling process (X, Y). Consider the stopped processes \(X_s^U=X_{s \wedge \tau _U}\) and \(Y_s^U=Y_{s \wedge \tau _U}\). In a slight abuse of notation, we use the same notation \(X_s^U\) for the coordinate representation for this stopped process in U, and similarly for \(Y_s^U\). Also we continue to write \(F_s\) for the coordinate representation of \(F_s: U \rightarrow V\).
Corollary 41
All the partial derivatives with respect to \(\mathbf {x}\) of \((s,\mathbf {x}) \mapsto F_s(\mathbf {x})\) are continuously differentiable in s. Furthermore, \(\left. \tfrac{{\text {d}}}{{\text {d}}s}\right _{s=t}F_s(\mathbf {x})\) is smooth in \(\mathbf {x}\).
Proof
3.6 Structure of the coupling
Lemma 42
Proof
Theorem 43
Remark 44
3.7 Classification of the drift
Finally it is possible to produce a complete characterization of the drift \(\mathbf {b}\) under LPC. Recall that M can only be a scaled version of one of the model spaces \(\mathbb {S}^d\), \(\mathbb {H}^d\) or \(\mathbb {R}^d\) corresponding to the curvature K being constant and equal to \(+1\), \(1\), or \(0\).
For this section, special attention is paid to the Eq. (71) at time 0. When the context makes it plain there is no ambiguity, we will write F for \(F_0\) and \(\kappa \) for \(\kappa _0\).
Let \(\nabla \) represent the covariant derivative with respect to the Riemannian connection compatible with the metric g. We will need the following useful fact about Killing vectorfields [32, Prop. 27].
Lemma 45
Isometries take geodesics to geodesics, so any Killing vectorfield is a Jacobi field, i.e. the variation field of a variation through geodesics. Thus, Killing vectorfields satisfy the Jacobi equation, as given by the following lemma [26, Theorem 10.2].
Lemma 46
Lemma 47
Proof
Recall that \(\mathbf {x}^*\) is the midpoint of a minimal geodesic connecting \(\mathbf {x}_0\) and \(\mathbf {y}_0\). Let \(\{e_1, \ldots , e_d\}\) denote the canonical orthonormal frame of \(T_{\mathbf {x}^*}M\). From previous discussions, F ‘inverts’ one geodesic through \(\mathbf {x}^*\) (the minimal geodesic joining \(\mathbf {x}_0\) and \(\mathbf {y}_0\)) and keeps all geodesics orthogonal to this one fixed. Let \(\mathbf {n} \in T_{\mathbf {x}^*}M\) denote the direction of the inverted geodesic.
Now we describe the drift vectorfield along geodesics issuing from \(\mathbf {x}^*\), the midpoint of a minimal geodesic joining \(\mathbf {x}_0\) and \(\mathbf {y}_0\). In the following, we will denote the canonical orthonormal basis of \(T_{\mathbf {x}^*}M\) by \(\{e_1,\ldots ,e_d\}\). Also, for any vector \(u \in T_{\mathbf {x}^*}M\) and any \(d \times d\) matrix T, Tu will denote the vector obtained by matrix multiplication when we identify \(T_{\mathbf {x}^*}M\) with \(\mathbb {R}^d\).
Lemma 48
Proof
Let \((n_t : t \ge 0)\) be the parallel transport of the vector normal to the hypersurface \(H(\mathbf {x},\mathbf {y})\) at \(\mathbf {x}^*\) along the geodesic \(\gamma \). Note that, as \(H(\mathbf {x},\mathbf {y})\) is totally geodesic, the second fundamental form vanishes identically on \(H(\mathbf {x},\mathbf {y})\) [26, Exercise 8.4]. This fact implies that parallel transportation of a vector \(v \in T_{\mathbf {x}^*}H(\mathbf {x},\mathbf {y})\) with respect to the induced metric on \(H(\mathbf {x},\mathbf {y})\) agrees with parallel transportation of v in the ambient manifold M [26, Lemma 8.5]. Thus, \(n_t\) is precisely the direction that is reversed at \(\gamma (t)\) by \(f_{\mathbf {x},\mathbf {y}}\).
Now, consider any parallel vectorfield \(V_t\) along \(\gamma \) which is orthogonal to \(\dot{\gamma }_t\). By the discussion following the definition of \(\mathcal {S}\), there exists a sequence of isometries \(\{F_k\}_{k \ge 1}\) such that each \(F_k\) is a composition of isometries in \(\mathcal {S}\), \(F_k\) fixes \(\gamma \), and \({\text {d}}F_k(n_0) \rightarrow V_0\) as \(k \rightarrow \infty \). As \(F_k\) fixes \(\mathbf {x}^*\) for each k, by [30, p. 7], we can choose a subsequence \(k_l\) such that \(F_{k_l} \rightarrow F\) in \(\mathcal {G}\) as \(l \rightarrow \infty \). Write \(V^{(k)}_t= F_{k*}n_t\). By [30, Lemma 4], for each \(t \ge 0\), \(V^{(k_l)}_t \rightarrow {\text {d}}F(n_t)\) in \(T_{\gamma (t)}M\) as \(l \rightarrow \infty \). In particular, \({\text {d}}F(n_0) = V_0\), and as F is an isometry fixing \(\gamma \), \({\text {d}}F(n_t) = V_t\) for all \(t \ge 0\). Thus, we have \(V^{(k_l)}_t \rightarrow V_t\) in \(T_{\gamma (t)}M\) for each \(t \ge 0\). From the discussion in the previous paragraph, (82) holds with \(V^{(k_l)}\) in place of V for each \(l \ge 1\). Taking \(l \rightarrow \infty \), we obtain (82) for the vectorfield V.
Finally, take any pair of unit vectors \(u, v \in T_{\mathbf {x}^*}M\) satisfying \(u \perp v\). Let \(\sigma \) be the geodesic issuing from \(\mathbf {x}^*\) such that \(\dot{\sigma }(0)=u\). We can obtain a sequence of isometries \(\{G_k\}_{k \ge 1}\) such that each \(G_k\) is a composition of isometries in \(\mathcal {S}\) and \({\text {d}}G_k(\dot{\gamma }(0)) \rightarrow u\) as \(k \rightarrow \infty \). Write \(u_k={\text {d}}G_k(\dot{\gamma }(0))\) and let \(\sigma _k\) be the geodesic issuing from \(\mathbf {x}^*\) in the direction \(u_k\). Denote by \(V^{v,k}_t\) and \(V^v_t\) the parallel transport of v along \(\sigma _k\) and \(\sigma \) respectively. By the previous discussion, we know that (82) holds with \(V^{v,k}\) in place of V and \(\sigma _k\) in place of \(\gamma \) for each \(k \ge 1\). Observe that for each fixed \(t \ge 0\), both sides of (82) depend continuously on u and v (this observation for the left hand side follows from the fact that the solution to the geodesic and parallel transport equations depends continuously on the initial data). Thus, we can take \(k \rightarrow \infty \) to get (82) with \(V^v\) in place of V and \(\sigma \) in place of \(\gamma \).
Lemma 49
\(K \ne 0\) implies \(\lambda =0\).
Proof
Denote by \(\sigma \) the geodesic issuing from \(\mathbf {x}_2\) and passing through \(\mathbf {x}_1\), and set \(\gamma \) to be a geodesic issuing from \(\mathbf {x}_2\) in a direction orthogonal to \(\sigma \). Locate \(\mathbf {z}=\gamma ({\text {dist}}(\mathbf {x}_1,\mathbf {x}_2))\). Taking \(\rho \) sufficiently small, we can ensure that \(\gamma \) restricted to \([0, {\text {dist}}(\mathbf {x}_1,\mathbf {x}_2)]\) is a minimal geodesic from \(\mathbf {x}_2\) to \(\mathbf {z}\). Finally, denote the geodesic issuing from \(\mathbf {x}_1\) and passing through \(\mathbf {z}\) by \(\eta \). Consider the geodesic triangle \(\Delta \) formed by \(\mathbf {x}_1\), \(\mathbf {x}_2\) and \(\mathbf {z}\). Thus, the sides of \(\Delta \) are formed by the geodesics \(\sigma \), \(\gamma \) and \(\eta \).
Finally we can state and prove the main theorem of this section.
Theorem 50
 (i)
The underlying Riemannian manifold M is one of the three model spaces \(\mathbb {S}^d\) \((K>0)\), \(\mathbb {R}^d\) \((K=0)\) or \(\mathbb {H}^d\) \((K<0)\), in the sense that the diffusion must be expressible as Riemannian Brownian motion plus drift vectorfield \(\mathbf {b}\) for such an \(M\).
 (ii)
For \(K \ne 0\), the drift \(\mathbf {b}\) must and can be any Killing vectorfield \(\mathcal {K}\) on M. For \(K=0\), the drift \(\mathbf {b}\) must and can be described in Euclidean coordinates by \(\mathbf {b}(\mathbf {x})=\lambda \mathbf {x}+T\mathbf {x}+ \mathbf {c}\) for any scalar \(\lambda \), any skewsymmetric matrix T and any vector \(\mathbf {c}\), where \(\mathbf {x}\mapsto \lambda \mathbf {x}\) is a dilation vectorfield about the origin and \(\mathbf {x}\mapsto T\mathbf {x}+ \mathbf {c}\) is a Killing vectorfield.
Proof
The classification of the space M is essentially the content of Theorem 38. Lemmas 48 and 49 show that if LPC holds then the drift vectorfield \(\mathbf {b}\) has to be of the form described in the theorem. For the case \(K=0\), Sect. 2 shows the existence of a Markovian maximal coupling with any pair of starting points \(\mathbf {x}\in \mathcal {B}(\mathbf {x}_0,r)\) and \(\mathbf {y}\in \mathcal {B}(\mathbf {y}_0,r)\) and fully describes the coupling.
Corollary 51
Under the hypothesis of part (ii) of Theorem 50, let \((\Upsilon _t : t \in \mathbb {R})\) denote the oneparameter subgroup of isometries corresponding to the Killing vectorfield \(\mathcal {K}\). Then for \(t \ge 0\), the mirror \(H_t\) and the corresponding reflection isometries \(F_t\) satisfy \(H_t=\Upsilon _t(H_0)\) and \(F_t=\Upsilon _t \circ F_0 \circ \Upsilon _t^{1}\).
Proof
Let \(Z, \widetilde{Z}\) be maximally coupled Brownian motions on M. For any \(t \ge 0\), by Remark 44, \(H_0=H(Z_t, \widetilde{Z}_t)\) almost surely. By Theorem 32, \(H_t=H(\Upsilon (Z_t), \Upsilon (\widetilde{Z}_t))\) almost surely. From this, \(H_t=\Upsilon _t(H_0)\) easily follows. Further, as \(F_t\) and \(\Upsilon _t \circ F_0 \circ \Upsilon _t^{1}\) have the same set of fixed points, namely \(H_t\), and neither of them is the identity, therefore \(F_t=\Upsilon _t \circ F_0 \circ \Upsilon _t^{1}\) follows from uniqueness of isometry with fixed point set \(H_t\). \(\square \)
In the following theorem, we characterise the class of drifts \(\mathbf {b}\) and starting points \(\mathbf {x}_0, \mathbf {y}_0\) for which the interface \(I(\mathbf {x}_0,\mathbf {y}_0,t)\) does not depend on time t.
Theorem 52
 (i)
\(K=0\), \(\mathbf {b}(\mathbf {x})=\lambda \mathbf {x}+T\mathbf {x}+ \mathbf {c}\) for some scalar \(\lambda \), skewsymmetric matrix T and vector \(\mathbf {c}\), and \(\mathbf {x}_0,\mathbf {y}_0,\lambda , T, \mathbf {c}\) satisfy \(T(\mathbf {x}_0\mathbf {y}_0)=0\) and \((\mathbf {x}_0\mathbf {y}_0)^\top (\lambda (\mathbf {x}_0+\mathbf {y}_0) + 2\mathbf {c})=0\).
 (ii)
\(K \ne 0\) and \(\mathbf {b}\) is a Killing vectorfield \(\mathcal {K}\) on M which satisfies the following: if \(\mathbf {x}^*\) is the midpoint of a minimal geodesic joining \(\mathbf {x}_0\) and \(\mathbf {y}_0\) and \(\mathbf {n}\) is the vector normal to the hypersurface \(H(\mathbf {x}_0,\mathbf {y}_0)\) at \(\mathbf {x}^*\), then \(\langle \mathcal {K}(\mathbf {x}^*), \mathbf {n}\rangle = 0\) and \(\nabla _{n}\mathcal {K}(\mathbf {x}^*)=0\).
Proof
When \(\lambda =0\), we get \(l(t)=l(0) + t (\mathbf {n}(0)^\top \mathbf {c})\). Thus \(l(t)=l(0)\) for all \(t \ge 0\) if and only if \((\mathbf {x}_0\mathbf {y}_0)^\top \mathbf {c}=0\).
Now, suppose \(K \ne 0\) and \(\mathbf {b}\) is the Killing vectorfield \(\mathcal {K}\) on M. As there is at most one isometry whose fixed point set is \(H(\mathbf {x}_0,\mathbf {y}_0)\), we deduce that \(I(\mathbf {x}_0,\mathbf {y}_0,t) = I(\mathbf {x}_0,\mathbf {y}_0,0)\) for all \(t \ge 0\) if and only if \(F_t = F\) for all \(t \ge 0\).
Conversely, suppose \(\langle \mathcal {K}(\mathbf {x}^*), \mathbf {n}\rangle = 0\) and \(\nabla _{n}\mathcal {K}(\mathbf {x}^*)=0\) holds. Let \(\gamma \) be any geodesic issuing from \(\mathbf {x}^*\) and lying in \(H(\mathbf {x}_0,\mathbf {y}_0)\) and let \(n_t\) denote the parallel transport of \(\mathbf {n}\) along \(\gamma \). As \(\langle \mathcal {K}(\mathbf {x}^*), \mathbf {n}\rangle = 0\) and \(\langle \nabla _{\dot{\gamma }(0)}\mathcal {K}(\mathbf {x}^*), \mathbf {n}\rangle = \langle \nabla _{\mathbf {n}}\mathcal {K}(\mathbf {x}^*), \dot{\gamma }(0)\rangle = 0\), using the representation (88) for \(\mathcal {K}\), we see that \(\langle \mathcal {K}(\gamma (t), n_t\rangle = 0\) and hence, \(\mathcal {K}(\gamma (t)) \in T_{\gamma (t)}H(\mathbf {x}_0,\mathbf {y}_0)\) for all \(t \ge 0\). As the submanifold \(H(\mathbf {x}_0,\mathbf {y}_0)\) is a geodesic space, we conclude that \(\mathcal {K}\) restricted to \(H(\mathbf {x}_0,\mathbf {y}_0)\) is a vectorfield tangent to this submanifold. Thus, if \(\Upsilon _t\) denotes the flow of isometries generated by \(\mathcal {K}\), then for each \(\mathbf {z}_0 \in H(\mathbf {x}_0,\mathbf {y}_0)\), \(\Upsilon _t(\mathbf {z}_0)\) lies in \(H(\mathbf {x}_0,\mathbf {y}_0)\) at least for a short time. As \(\Upsilon _t\) is a global flow (because M is complete), a routine compactness argument implies that \(\Upsilon _t(\mathbf {z}_0) \in H(\mathbf {x}_0,\mathbf {y}_0)\) for all \(t \ge 0\). Thus, by Corollary 51, \(H_t \subseteq H(\mathbf {x}_0,\mathbf {y}_0)\), and hence \(F_t = F\), for all \(t \ge 0\). \(\square \)
4 Conclusion
In this paper we have shown that Markovian maximal couplings of regular elliptic diffusions with smooth coefficients (and satisfying diffusiongeodesic completeness and stochastic completeness) have to be reflection couplings tied to involutive isometries of the corresponding Riemannian structure on state space; moreover as soon as the existence of a Markovian maximal coupling is stable (in the sense of LPC) then a rigidity result requires the Riemannian structure to be Euclidean, hyperspherical, or hyperbolic, and the space must be simply connected. In such cases the drift must also be of a very simple form, corresponding to a rotation with possibly (but only in the Euclidean case) a dilation component.
Thus Markovian maximal couplings of elliptic diffusions are rare, and their existence enforces severe geometric constraints.
It is natural to ask whether the assumptions of diffusiongeodesic completeness and stochastic completeness are required. It seems likely that they are not required, but (this paper already being long) we save this question for another occasion.
The scarcity of Markovian maximal couplings places a natural premium on questions of efficiency of Markovian coupling, as discussed for example in [6], for the case of reflecting Brownian motion in compact regions. One could ask, for example, when it is possible to construct Markovian couplings \((X, Y)\) which are optimal in the sense that the tail probability of the coupling time \(\mathbb {P}\left[ \tau >t\right] \) is minimized for all \(t\) amongst Markovian couplings if not amongst all possible couplings. (Note that this notion of optimality differs from the optimality discussed in [8], which is defined relative to a specified Wasserstein metric.) Little is known as yet about such couplings, though [22] exhibits a coupling of two copies of scalar Brownian motion and local time which is Markovian, nonmaximal, but optimal amongst all Markovian couplings. The question of whether similar geometric rigidity results for existence of such optimal Markovian couplings remains entirely open, and its answer would be of great interest.
We expect that in fact such optimal Markovian couplings are also rare. Further refinements are possible (for example, one could consider the existence of Markovian couplings which minimize the Laplace transform \(\mathbb {E}\left[ \exp \left( u\tau \right) \right] \) for some or all values of \(u>0\)); however the probable rarity of such couplings would focus attention on developing the notions of efficiency from [6] to apply to noncompact regions. In particular there is a natural question concerning criteria for existence of efficient Markovian couplings, where “efficient” here means, the rate of decay of \(\mathbb {P}\left[ \tau >t\right] \) with \(t\) for the Markovian coupling is comparable to that of the total variation distance \(\Vert \mu _{1,t}\mu _{2,t}\Vert _{TV}\) between the onepoint distributions \(\mu _{1,t}\) and \(\mu _{2,t}\) (the distributions of \(X_t\) and \(Y_t\) respectively).
 1.
extension of the notion of Markovian maximal coupling to the hypoelliptic case (in which case in fact the very existence of Markovian couplings is moot: but see the positive results of [21, 23]);
 2.
examination of the extent to which the ideas of this paper carry over to Markov processes which are not skipfree (and here a natural first step would be to consider the case of couplings of Lévy processes, though a potentially significant result in the random walk case is to be found in [35]).
Notes
Acknowledgments
We wish to thank an anonymous referee whose very careful reading of the manuscript and detailed comments greatly improved the article.
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