Coupling of Brownian motions and Perelman s L-functional

Transcription

1 Coupling of Brownian motions and Perelman s L-functional Kazumasa Kuwada and Robert Philipowski Abstract We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that their normalized L-distance is a supermartingale. As a corollary, we obtain the monotonicity of the transportation cost between two solutions of the heat equation in the case that the cost function is the composition of a concave non-decreasing function and the normalized L-distance. In particular, it provides a new proof of a recent result of Topping J. reine angew. Math , Keywords: Ricci flow, L-functional, Brownian motion, coupling. AMS subject classification: 53C44, 58J65, 6J65. 1 Introduction Let M be a d-dimensional differentiable manifold, τ 1 < τ 2 < T and gτ τ τ1,t a complete backwards Ricci flow on M, i.e. a smooth family of Riemannian metrics satisfying g τ = 2 Ric gτ 1 and such that M, gτ is complete for all τ τ 1, T. In this situation Perelman 24, Section 7.1 see also 6, Definition 7.5 defined the L-functional of a smooth curve γ : τ 1, τ 2 M where τ 1 τ 1 < τ 2 T by τ2 Lγ := τ γτ 2 gτ + R gτγτ dτ, τ 1 where R gτ x is the scalar curvature at x with respect to the metric gτ. Let Lx, τ 1 ; y, τ 2 be the L-distance between x, τ 1 and y, τ 2 defined by the infimum of Lγ over smooth curves γ : τ 1, τ 2 M satisfying γτ 1 = x and γτ 2 = y. The normalized L-distance Θ t x, y t 1 is given by Θ t x, y := 2 τ 2 t τ 1 t Lx, τ 1 t; y, τ 2 t 2d τ 2 t τ 1 t 2. For a measurable function c : M M R, let us define the transportation cost T c µ, ν between two probability measures µ and ν on M with respect to the cost function c by T c µ, ν := inf cx, yπdx, dy π M M the infimum is over all probability measures π on M M whose marginals are µ and ν respectively. To study Perelman s work from a different aspect, Topping 31 see also Lott 2 showed the following result: Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo , Japan. kuwada.kazumasa@ocha.ac.jp Partially supported by the JSPS fellowship for research abroad. Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 6, Bonn, Germany. philipowski@iam.uni-bonn.de 1

2 Theorem 1 Theorem 1.1 in 31. Assume that M is compact and that τ 1 >. Let p : τ 1, T M R + and q : τ 2, T M R + be two non-negative unit-mass solutions of the heat equation p τ = gτp Rp, where the term Rp comes from the change in time of the volume element. Then the normalized L-transportation cost T Θt p τ 1 t, vol g τ1 t, q τ 2 t, vol g τ2 t between the two solutions evaluated at times τ 1 t resp. τ 2 t is a non-increasing function of t 1, T/ τ 2. By gτ-brownian motion, we mean the time-inhomogeneous diffusion process whose generator is gτ. As in the time-homogeneous case, the heat distribution pτ, vol gτ is expressed as the law of a gτ-brownian motion at time τ. In view of this strong relation between heat equation and Brownian motion, it is natural to ask whether one can couple two Brownian motions on M in such a way that a pathwise analogue of this result holds. The main result of this paper answers it affirmatively as follows: Theorem 2. Assume that the Ricci curvature of M is bounded from below uniformly, namely there exists K such that Ric gτ Kgτ for any τ τ 1, T. 2 Then given any points x, y M and any s 1, T/ τ 2, there exist two coupled gτ-brownian motions X τ τ τ1 s,t and Y τ τ τ2 s,t with initial values X τ1 s = x and Y τ2 s = y such that the process Θ t X τ1 t, Y τ2 t t s,t/ τ2 is a supermartingale. In addition, we can take them so that the map x, y X, Y is measurable. In particular, for any ϕ : R R being concave and non-decreasing, E ϕθ t X τ1 t, Y τ2 t is non-increasing. Obviously, 2 is satisfied if M is compact. Thus it includes the case of Theorem 1. As a result, Theorem 2 easily implies the following extension of Theorem 1 see subsection 5.3: Theorem 3. Assume that 2 holds. Let ϕ : R R be concave and non-decreasing. Then T ϕ Θt p τ 1 t, vol g τ1 t, q τ 2 t, vol g τ2 t is non-increasing in t 1, T/ τ 2 for non-negative unit mass solutions p and q to the heat equation. We prove Theorem 2 by constructing a coupling via approximation of gτ-brownian motions by geodesic random walks as studied in 15. In the next section, we demonstrate background of the problem, review related results and compare their methods with ours. Since our method superficially looks like a detour compared with other existing coupling arguments, there we explain the reason why we choose that way. To state the idea behind our proof explicitly, we prove Theorem 2 under the assumption that there is no singularity of L-distance in section 3. Since all technical difficulties are concentrated on the singularity of L-distance, we can study the problem there in more direct way by using stochastic calculus. Some estimates on variations of L-distance is gathered in section 4. The proof of the full statement of Theorems 2 and 3 will be provided in section 5. Before closing this section, we give two remarks on Theorems 2 and 3. Remark 1. As shown in 17, under backwards Ricci flow gτ-brownian motion cannot explode. Hence Θ t X t, Y t is well-defined for all t s, T/ τ 2 in Theorem 2. This fact also ensures that pτ, vol gτ has unit mass whenever it does at the initial time. We implicitly require this property to make T ϕ Θt p τ 1 t, vol g τ1 t, q τ 2 t, vol g τ2 t well-defined in Theorem 3. 2

3 Remark 2. There are plenty of examples of backwards Ricci flow satisfying 2 even when M is non-compact. Indeed, given a metric g on M with bounded curvature tensor, there exists a unique solution to the Ricci flow t gt = 2 Ric gt with initial condition g satisfying sup Rm gt gt x < x,t for a short time see 29 for existence and 5 for uniqueness. Then the corresponding backwards Ricci flow satisfying 2 is obtained by time-reversal. 2 Review and remarks on background of the problem The Ricci flow was introduced by Hamilton 1. There he effectively used it to solve the Poincaré conjecture for 3-manifolds with positive Ricci curvature. By following his approach, Perelman 24, 25, 26 finally solved the Poincaré conjecture see also 4, 13, 23. There he used L- functional as a crucial tool. At the same stage, he also studied the heat equation in 24 in relation with the geometry of Ricci flows. It suggests that analysing the heat equation is still an efficient way to investigate geometry of the underlying space even in the time-dependent metric case. This guiding principle has been confirmed in recent developments in this direction. For example, we refer to 37 as one of such developments. In connection with the theory of optimal transportation, McCann and Topping 22 showed the monotonicity of T ρ 2 pτ, vol gτ gτ, qτ, vol gτ, where ρ gτ is the gτ-riemannian distance, under backwards Ricci flow on a compact manifold. Topping s result 31 can be regarded as an extension of it to contraction in the normalized L- transportation cost see 2 also. By taking τ 2 τ 1, he gave a new proof of the monotonicity of Perelman s W-entropy, which is one of fundamental ingredients in Perelman s work. A probabilistic approach to these problems is initiated by Arnaudon, Coulibaly and Thalmaier. In 2, Section 4, they sharpened McCann and Topping s result 22 to a pathwise contraction in the following sense: There is a coupling X t, Y t t of two Brownian motions starting from x, y X respectively such that the gt-distance between X t and Y t is non-increasing in t almost surely. In their approach, probabilistic techniques based on analysis of sample paths made it possible to establish such a pathwise estimate. As an advantage of their result, the pathwise contraction easily yields that T ϕ ρgτ pτ, vol gτ, qτ, vol gτ is non-increasing for any non-decreasing ϕ. As an application of this sharper monotonicity, we can obtain an L 1 -gradient estimate of Bakry-Émery type see 16 for the heat semigroup. In the time-homogeneous case, this gradient estimate has been known to be very useful in geometric analysis see e.g. 3, 18. McCann and Topping s result only implies L 2 -gradient estimate and it is weaker than the L 1 - estimate In the time-homogeneous case, it is known that L 2 -estimate also implies L 1 -estimate see 3, 18 and 27. However, to the best of our knowledge, an extension of such equivalence in the time-inhomogeneous case is not yet established. As another advantage of Arnaudon, Coulibaly and Thalmaier s approach, their argument works even on non-compact M cf. 17. Our theorem 2 can be regarded as an extension of their result. Indeed, our approach is the same as theirs in spirit and advantages of probabilistic approach as mentioned are also inherited to our results as we have seen in Theorem 3. We can expect that our approach makes it possible to employ several techniques in stochastic analysis to obtain more detailed behavior of Θ t X τ1 t, Y τ2 t, especially in the limit τ 2 τ 1, in a future development. Now we compare our method of the proof with existing arguments in coupling methods from a technical point of view. We hope that the following observation would be helpful to extend other coupling arguments than ours in this case. A common and basic idea is to couple infinitesimal motions of two Brownian particles by using a parallel transport of tangent vectors along a minimal geodesic joining them. Thus the technical difficulty arises from the singularity 3

4 of L-distance, or the presence of L-cut locus. In our approach, we consider coupled geodesic random walks each of which approximates the Brownian motion. After we establish a difference inequality for time evolution of the L-distance between coupled random walks, we will obtain the result by taking a limit. Note that the convergence of our random walk in law to the Brownian motion in this time-inhomogeneous case is already established in 15. In the time-homogeneous case, there are several arguments 9, 11, 33, 34, 35 to construct such a coupling by approximating it with ones which move as mentioned above if they are distant from the cut locus and move independently if they are close to the cut locus. In these cases, it will be important to estimate the size of the total time when particles are close to the cut locus. To do the same in the time-inhomogeneous case, it does not seem straightforward since the L-cut locus depends on time, namely it moves as time evolves. In our approach, instead of applying stochastic calculus, we only need to show a difference inequality. Thus the singularity at the L-cut locus causes less difficulties at this stage see Remark 7 for more details. If we employ the theory of optimal transportation, we will work on coupling of heat distributions instead of coupling of Brownian motions. Once we move to the world of heat distributions, we can ignore the cut locus since they are of measure zero with respect to the Riemannian measure. However, in the derivation of the monotonicity results, the theory of optimal transportation at present covers only the case that the cost function is squared distance or L-distance. It reflects the difference of results between McCann and Topping 22 and Arnaudon, Coulibaly and Thalmaier 2. It should be remarked that such a difference still exists between these two approaches, the one by optimal transportation and the other by stochastic analysis, even in the time-homogeneous case. Arnaudon, Coulibaly and Thalmaier 2 study the problem by developing a new method. They constructed one-parameter family of coupled particles X t u t u,1 so that X t u moves as a Brownian motion for any u and X t u u,1 is a C 1 -curve whose length is nonincreasing in t. Thus X t, X t 1 is the expected coupling. To construct it, they first consider a finite number of particles X t u i t i which are coupled with other particles by parallel transport. Then, by increasing the number of particles, we obtain such a one-parameter family in the limit. Since they are coupled by parallel transport, the distance between two particles is of bounded variation at least before they hit the cut locus. Thus, if adjacent particles are sufficiently close to each other at time t, we can take a deterministic δ > such that they cannot hit the cut locus at least until time t + δ. Based on this observation, they succeeded in avoiding the problem coming from the cut locus by increasing the number of particles to make it constitute a one-parameter family of particles. In the case of this paper, we work on the L-distance instead of the Riemannian distance and construct a coupling by space-time parallel transport instead of a coupling by parallel transport. As a result, L-distance between coupled particle is not of bounded variation see Remark 6 for more details. Thus, our problem differs in nature from what is studied in 1. If we want to extend Arnaudon, Coulibaly and Thalmaier s approach in the present case, we have to be careful and need some additional arguments. Even if we succeed in constructing a one-parameter family of particles X t u t u,1 coupled by space-time parallel transport, we cannot expect that X t u u,1 is a C 1 -curve. In our approach, such a difference causes no additional difficulty. Indeed, as studied in 14, 15, we already know that it works to construct coupled particles by reflection, the distance of which naturally regarded as a semimartingale with a non-vanishing martingale part. 3 Coupling of Brownian motions in the absence of L-cut locus Since the proof of Theorem 2 involves some technical arguments, first we study the problem in the case that the L-distance L has no singularity. More precisely, we do it here under the 4

5 following assumption: Assumption 1. The L-cut locus is empty. See subsection 5.1 or 6, 31, 36 for the definition of L-cut locus. Under Assumption 1, the following holds: 1. For all x, y M and all τ 1 τ 1 < τ 2 T there is a unique minimizer γ τ 1τ 2 xy of Lx, τ 1 ; y, τ 2 existence of γ τ 1τ 2 xy is proved in 6, Lemma 7.27, while uniqueness follows immediately from the characterization of L-cut locus, see subsection The function L is globally smooth. Thus, in this case, we can freely use stochastic analysis on the frame bundle without taking any care on regularity of L. 3.1 Construction of the coupling A gτ-brownian motion X on M scaled in time by the factor τ 1 starting at a point x M at time s 1, T/ τ 2 can be constructed in the following way 1, 8, 17: Let π : FM M be the frame bundle and e i d the standard basis of Rd. For each τ τ 1, T let H i τ d be the associated gτ-horizontal vector fields on FM i.e. H i τ, u is the gτ-horizontal lift of ue i. Moreover let V α,β d α,β=1 be the canonical vertical vector fields, i.e. Vα,β fu := m αβ m=id fum m = m αβ d α,β=1 GL dr, and let W t t be a standard R d -valued Brownian motion. By O gτ M, we denote the gτ-orthonormal frame bundle. We first define a scaled horizontal Brownian motion as the solution Ũ = Ũt t s,t/ τ1 of the Stratonovich SDE dũt = 2 τ 1 H i τ 1 t, Ũt dwt i τ 1 α,β=1 g τ τ 1tŨte α, Ũte β V αβ Ũtdt 3 on FM with initial value Ũs = u O g τ 1s x M, and then define a scaled Brownian motion X on M as X t := πũt. Note that X t does not move when τ 1 =. The last term in 3 ensures that Ũt O g τ 1t M for all t s, T/ τ 1 see 1, Proposition 1.1, 8, Proposition 1.2, so that by Itô s formula for all smooth f : s, T/ τ 1 M R dft, X t = f t t, X t dt + 2 τ 1 Ũte i ft, X t dwt i + τ 1 gτ1 tft, X t dt. Let us define X τ τ τ1 s,t by X τ1 t := X t. Then X τ becomes a gτ-brownian motion when τ 1 >. Remark 3. Intuitively, it might be helpful to think that X τ lives in M, gτ, or X t lives in M, g τ 1 t. The same is true for Y and Ỹ which will be defined below. Similarly, for all curves γ : τ 1, τ 2 M appearing in connection with L-distance, we can naturally regard γτ as in M, gτ. We now want to construct a second scaled Brownian motion Ỹ on M in such a way that its infinitesimal increments dỹt are space-time parallel to those of X up to scaling effect along the minimal L-geodesic namely, the minimizer of L from X t, τ 1 t to Ỹt, τ 2 t. To make this idea precise, we first define the notion of space-time parallel vector field: 5

6 Definition 1 space-time parallel vector field. Let τ 1 τ 1 < τ 2 T and γ : τ 1, τ 2 M be a smooth curve. We say that a vector field Z along γ is space-time parallel if gτ γτ Zτ = Ric# gτ Zτ 4 holds for all τ τ 1, τ 2. Here gτ stands for the covariant derivative associated with the gτ-levi-civita connection and Ric # gτ is defined by regarding the gτ-ricci curvature as a 1,1-tensor via gτ. Since 4 is a linear first-order ODE, for any ξ T γτ1 M there exists a unique space-time parallel vector field Z along γ with Zτ 1 = ξ. Remark 4. Whenever Z and Z are space-time parallel vector fields along a curve γ, their gτ-inner product is constant in τ: d dτ Zτ, Z τ gτ = g τ τzτ, Z τ + gτ γτ Zτ, Z τ gτ + Zτ, gτ γτ Z τ gτ = 2 Ric gτ Zτ, Z τ Ric gτ Zτ, Z τ Ric gτ Zτ, Z τ =. Definition 2 space-time parallel transport. For x, y M and τ 1 τ 1 < τ 2 T, we define a map m τ 1τ 2 xy : T x M T y M as follows: m τ 1τ 2 xy ξ := Zτ 2, where Z is the unique space-time parallel vector field along γ τ 1τ 2 xy with Zτ 1 = ξ. By Remark 4, m τ 1τ 2 xy is an isometry from T x M, gτ 1 to T y M, gτ 2. In addition, it smoothly depends on x, τ 1, y, τ 2 under Assumption 1. Remark 5. The emergence of the Ricci curvature in 4 is based on the Ricci flow equation 1. Indeed, we can introduce the notion of space-time parallel transport even in the absence of 1 with keeping the property in Remark 4 by using 2 1 τ gτ # instead of Ric # gτ in 4. This would be a natural extension in the sense that it coincides with the usual parallel transport when gτ is constant in τ. Similarly as in 11, Formula 6.5.1, we now define a second scaled horizontal Brownian motion Ṽ = Ṽt t s,t/ τ2 on FM as the solution of dṽt = 2 τ 2 H i τ 2 t, Ṽt dbt i τ 2 db t = Ṽ 1 t m τ 1,τ 2 πũt,πṽt Ũ t dw t α,β=1 g τ τ 2tṼte α, Ṽte β V αβ Ṽtdt with initial value Ṽs = v O g τ 2s y M, and we set Ỹt := πṽt. As we did for X, let us define Y τ τ τ2 s,t by Y τ2 t := Ỹt to make Y a gτ-brownian motion. From a theoretical point of view, it seems to be natural to work with X τ, Y τ see Remark 3. However, for technical simplicity, we will prefer to work with X t, Ỹt instead in the sequel. 3.2 Proof of Theorem 2 in the absence of L-cut locus Our argument in this section is based on the following Itô formula for X t, Ỹt: Lemma 1. Let f be a smooth function on s, T/ τ 2 M M. Then dft, X t, Ỹt = f t t, X t, Ỹtdt τ1 Ũ t e i 2 τ 2 Ṽ t e i ft, X t, ỸtdW i t Hess g τ1 t g τ 2 t f τ1 t, Xt,Ỹt Ũ t e i τ 2 Ṽ t e i, τ 1 Ũ t e i τ 2 Ṽ t e i dt. 6

7 Here the Hessian of f is taken with respect to the product metric g τ 1 t g τ 2 t, e i stands for e i Ũt, τ 1 t; Ṽt, τ 2 t, where e i u, τ 1 ; v, τ 2 := v 1 m τ 1,τ 2 πu,πvue i, and for tangent vectors ξ 1 T x M, ξ 2 T y M we write ξ 1 ξ 2 := ξ 1, ξ 2 T x,y M M. Proof. As in 12, Formula 2.11, Itô s formula applied to a smooth function f on s, T/ τ 2 FM FM gives d ft, Ũt, Ṽt = f t, Ũt, Ṽtdt t + 2 τ1 H i,1 τ 1 t, f t, Ũt, ṼtdWt i + 2 τ 2 H i,2 τ 2 t, f t, Ũt, ṼtdBt i + τ 1 Hi,1 τ 2 1 t, f t, Ũt, Ṽt + τ 2 Hi,2 τ 2 2 t, f t, Ũt, Ṽt dt + 2 τ 1 τ 2 α,β=1 i,j=1 H i,1 τ 1 t, H j,2 τ 2 t, f t, Ũt, Ṽtd W i, B j t g τ 1 τ τ 1tŨte α, Ũte β V αβ g Ũt τ 2 τ τ 2tṼte α, Ṽte β V αβ Ṽt ft, Ũt, Ṽtdt, where H i,1 resp. H i,2 means H i applied with respect to the first resp. second space variable. By the definition of B, this equals f t t, Ũt, Ṽtdt + + α,β=1 2 τ1 H i τ 1 t, Ũt 2 τ 2 Hi Ũt, τ 1 t; Ṽt, τ 2 t ft, Ũ t, ṼtdWt i τ1 H i τ 1 t, Ũt τ 2 2 Hi Ũt, τ 1 t; Ṽt, τ 2 t ft, Ũ t, Ṽtdt g τ 1 τ τ 1tŨte α, Ũte β V αβ g Ũt τ 2 τ τ 2tṼte α, Ṽte β V αβ Ṽt ft, Ũt, Ṽtdt, where H i u, τ 1; v, τ 2 is the gτ 2 -horizontal lift of ve i u, τ 1; v, τ 2. The claim follows by choosing ft, u, v := ft, πu, πv because this f is constant in the vertical direction so that the term involving V αβ f vanishes. Let Λt, x, y := Lx, τ 1 t; y, τ 2 t. In order to apply Lemma 1 to the function Θ we need the following proposition, whose proof is given in the next section. Since we will use it again in section 5, we state it without assuming Assumption 1. Proposition 1. Take x, y M, u O g τ 1t x M and v O g τ 2t y M. Let γ be a minimizer of Lx, τ 1 t; y, τ 2 t. Assume that x, τ 1 t; y, τ 2 t is not in the L-cut locus. Set ξ i := τ 1 ue i 7

8 τ2 ve i u, τ 1t; v, τ 2 t. Then τ2 t Λ t t, x, y = 1 3 τ 3/2 t τ 1 t 2τ R gτγτ gτ R gτ γτ 2 Ric gτ 2 gτ γτ 1 2τ γτ 2 gτ + 2 Ric gτ γτ, γτ dτ, 5 Hess gτ1 gτ 2 Λ ξ i, ξ i t,x,y d τ t and consequently τ= τ 2 t + 1 τ= τ 1 t t Λ t t, x, y+ τ2 t τ 1 t τ 3/2 2 Ric gτ 2 gτ γτ + gτr gτ γτ 2 τ R gτγτ 2 Ric gτ γτ, γτ dτ 6 Hess gτ1 gτ 2 Λ t,x,y ξ i, ξ i d t τ2 τ 1 1 2t τ2 t τ 1 t = d τ2 τ 1 1 Λt, x, y. t 2t τ R gτ γτ + γτ 2 gτ dτ The proof of Theorem 2 is now achieved under Assumption 1 by combining Lemma 1 and Proposition 1: Proof of Theorem 2 under Assumption 1. Since Θ is bounded from below, it suffices to show that the bounded variation part of Θ t X t, Ỹt is non-positive. By Lemma 1, dθ t X t, Ỹt = t Θ t X t, Ỹt + + Hess g τ1 t g τ 2 t Θ t τ1 Xt,Ỹt Ũ t e i τ 2 Ṽ t e i, τ 1 Ũ t e i τ 2 Ṽ t e i dt For the bounded variation part we obtain 2 τ1 Ũ t e i 2 τ 2 Ṽ t e i Θ t X t, ỸtdW i t. t Θ t X t, Ỹt = τ2 τ 1 Λt, X t, Ỹt + 2 τ 2 t τ 1 t Λ t t t, X t, Ỹt 2d τ 2 τ 2 1 and Hess g τ1 t g τ 2 t Θ t τ1 Xt,Ỹt Ũ t e i τ 2 Ṽ t e i, τ 1 Ũ t e i τ 2 Ṽ t e i = 2 τ 2 t τ 1 t d Hess g τ1 t g τ 2 t Λ τ1 t, Xt,Ỹt Ũ t e i τ 2 Ṽ t e i, τ 1 Ũ t e i τ 2 Ṽ t e i. 8

9 Thus, by Proposition 1, t Θ t X t, Ỹt + Hess g τ1 t g τ 2 t Θ t τ1 Xt,Ỹt Ũ t e i τ 2 Ṽ t e i, τ 1 Ũ t e i τ 2 Ṽ t e i 2 τ 2 t τ 1 t d t τ2 τ 1 1 2t Λt, X t, Ỹt τ2 τ 1 + Λt, X t, Ỹt 2d τ 2 τ 2 1 t =. Hence Θ t X t, Ỹt is indeed a supermartingale. Remark 6. Unlike the case in 1, the pathwise contraction of Θ t X t, Ỹt is no longer true in our case. In other words, the martingale part of Θ t X t, Ỹt does not vanish. We will see it in the following. The minimal L-geodesic γ = γ τ 1τ 2 xy of Lx, τ 1 ; y, τ 2 satisfies the L-geodesic equation gτ γτ γτ = 1 2 gτ R gτ 2 Ric # gτ γτ 1 γτ 7 2τ see 6, Corollary Thus the first variation formula see 6, Lemma 7.15 yields 2 τ1 Ũ t e i 2 τ 2 Ṽ t e i Λt, X t, Ỹt = 2t τ 2 Ṽte i, γ τ 2 t g τ2 t 2t τ 1 Ũte i, γ τ 1 t g τ1 t. 8 One obstruction to pathwise contraction is in the difference of time-scalings τ 1 and τ 2. addition, by 7, τ γτ is not space-time parallel to γ in general cf. Remark 4. In 4 Proof of Proposition 1 In this section, we write τ 1 := τ 1 t and τ 2 := τ 2 t. We assume τ 2 < T. For simplicity of notations, we abbreviate the dependency on the metric gτ of several geometric quantities such as Ric, R, the inner product,, the covariant derivative etc. when our choice of τ is obvious. For this abbreviation, we will think that γτ is in M, gτ and γτ is in T γτ M, gτ. Note that, when τ 1 =, lim τ τ1 τ γτ exists while limτ γτ =. In any case, τ γτ is locally bounded see Lemma 3. We first compute the time derivative of Λ. When τ 1 >, by 31, Formulas A.4 and A.5 we have L τ 1 x, τ 1 ; y, τ 2 = τ 1 Rgτ1 x γτ 1 2, L τ 2 x, τ 1 ; y, τ 2 = τ 2 Rgτ2 y γτ 2 2, so that Λ t t, x, y = τ L L 1 x, τ 1 ; y, τ 2 + τ 2 x, τ 1 ; y, τ 2 τ 1 τ 2 = 1 τ 3/2 2 Rγτ2 γτ 2 2 τ 3/2 1 Rγτ1 γτ t 9

10 Thus the integration-by-parts formula yields, Λ t t, x, y = 3 2t + 1 t τ2 τ 1 τ2 τ Rγτ γτ 2 dτ τ 1 τ 3/2 R τ γτ + γτrγτ 2 γτ γτ, γτ 2 Ric γτ, γτ dτ. 1 Note that we have R τ = R 2 Ric 2 11 see e.g. 3, Proposition Since γ satisfies the L-geodesic equation 7, by substituting 7 and 11 into 1, we obtain 5. Note that the derivation of 9 and 1 is still valid even when τ 1 = because of the remark at the beginning of this section. Thus 5 holds when τ 1 =, too. In order to estimate d Hess gτ 1 gτ 2 Λ t,x,y ξ i, ξ i we begin with the second variation formula for the L-functional: Lemma 2 Second variation formula; 6, Lemma Let Γ : ε, ε τ 1, τ 2 M be a variation of γ, Ss, τ := s Γs, τ, and Zτ := s Γ, τ the variation field of Γ. Then d 2 ds 2 LΓ s = 2 τ γτ, Zτ S, τ τ=τ 2 2 τ RicZτ, Zτ τ=τ 2 τ=τ 1 τ=τ 1 s= + 1 τ Zτ 2 τ=τ 2 τ2 + 2 τ τ 1 τ2 τ=τ 1 τ 1 τh γτ, Zτdτ γτzτ + Ric # Zτ 1 2τ Zτ 2 dτ, 12 where H γτ, Zτ := 2 Ric τ Zτ, Zτ Hess RZτ, Zτ + 2 Ric# Zτ 2 1 RicZτ, Zτ 2 RmZτ, γτ, γτ, Zτ τ 4 γτ RicZτ, Zτ + 4 Zτ Ric γτ, Zτ. 13 In 6 this lemma is only proved in the case τ 1 = and Zτ 1 =. However, the proof given there can be easily adapted to the slightly more general case needed here. Corollary 1 see 6, Lemma 7.39 for a similar statement. If the variation field Z is of the form τ Zτ = t Z τ 14 with a space-time parallel field Z satisfying Z τ 1, then d 2 ds 2 LΓ s = 2 τ γτ, Zτ S, τ gτ τ=τ 2 2 τ RicZτ, Zτ τ=τ 2 τ=τ 1 τ=τ 1 s= τ2 τ τ=τ 2 τh γτ, Zτdτ + τ 1 t τ=τ 1. 1

11 Proof. Since Z is space-time parallel, Z satisfies so that the last term in 12 vanishes. γτ Zτ = Ric # Zτ + 1 Zτ, 15 2τ Corollary 2 Hessian of L; see 6, Corollary 7.4 for a similar statement. Let Z be a vector field along γ of the form 14 and ξ := Zτ 1 Zτ 2 T x,y M M. Then Hess gτ1 gτ 2 L τ2 x,τ1 ;y,τ 2 ξ, ξ τ τ=τ 2 τh γτ, Zτdτ + τ 1 t τ=τ 1 2 τ Ric gτ Zτ, Zτ τ=τ 2 τ=τ Proof. Let Γ : ε, ε τ 1, τ 2 M be any variation of γ with variation field Z and such that Zτ1 S, τ 1 and Zτ2 S, τ 2 vanish. 17 Let ls := LΓs, τ 1, τ 1 ; Γs, τ 2, τ 2 and ˆls := LΓs,. Since ˆl = l and ˆls ls for all s ε, ε, we have l ˆl so that, using 17, Hess gτ1 gτ 2 L d2 x,τ1 ξ, ξ = ;y,τ 2 ds 2 LΓs, τ 1, τ 1 ; Γs, τ 2, τ 2 s= = l ˆl = d2 ds 2 LΓ s. s= The claim now follows from Corollary 1. Let now Zi i = 1,..., d be space-time parallel fields along γ satisfying Zi τ 1 = ue i and consequently Zi τ 2 = ve i, and Z iτ := τ/tzi τ so that ξ i = Z i τ 1 Z i τ 2. In order to estimate d Hess gτ 1 gτ 2 L x,τ1 ;y,τ 2 ξ i, ξ i using Corollary 2 we will compute d H γτ, Z iτ in the following see 6, Section for a similar argument. Set I 1, I 2 and I 3 by I 1 := 2 I 2 := I 3 := 4 Ric τ Z iτ, Z i τ, Hess RZ i τ, Z i τ + 2 Ric # Z i τ 2 1 τ RicZ iτ, Z i τ 2 RmZ i τ, γτ, γτ, Z i τ, Zi τ RicZ i τ, γτ γτ RicZ i τ, Z i τ. Then d H γτ, Z iτ = I 1 + I 2 + I 3 holds. By a direct computation, I 2 = τ Rγτ + 2 Ric 2 γτ 1τ t Rγτ + 2 Ric γτ, γτ. 18 The contracted Bianchi identity div Ric = 1 2 R 19, Lemma 7.7 yields I 3 = 4τ t div Ric γτ γτ Rγτ = 2τ t γτrγτ

12 For I 1, we have I 1 = 2 = 2 d dτ = 2τ t Since Z i satisfies 15, 4 d dτ RicZ iτ, Z i τ γτ RicZ i τ, Z i τ 2 Ric γτ Z i τ, Z i τ τ t Rγτ + 2 τ t γτrγτ R Rγτ + τ τ γτ + 4 Ric γτ Z i τ, Z i τ Ric γτ Z i τ, Z i τ. 2 Ric γτ Z i τ, Z i τ = 4 Ric Ric # Z i τ + 1 2τ Z iτ, Z i τ = 2τ 2 Ric 2 γτ 1τ t Rγτ. 21 By substituting 21 into 2, I 1 = 2τ t R τ γτ + 2 Ric 2 γτ. 22 Hence, by combining 22, 19 and 18, H γτ, Z i τ = τ t Inserting this into 16 we obtain Hess gτ1 gτ 2 L x,τ1 ;y,τ 2 ξ i, ξ i τ2 2 R τ γτ 2 Ric 2 γτ Rγτ 1 τ Rγτ + 2 Ric γτ, γτ 2 γτrγτ. 1 τ 3/2 2 R t τ 1 τ γτ + 2 Ric 2 γτ + Rγτ + 1 τ Rγτ 2 Ric γτ, γτ + 2 γτrγτ dτ + d τ τ=τ 2 t 2τ 3/2 τ=τ 2 Rγτ τ=τ 1 t τ=τ 1 = d τ τ=τ 2 t + 1 τ2 τ 2 3/2 Ric 2 γτ + Rγτ 2 τ=τ 1 t τ 1 τ Rγτ 2 Ric γτ, γτ dτ which completes the proof of Proposition 1. 5 Coupling via approximation by geodesic random walks To avoid a technical difficulty coming from singularity of L on the L-cut locus, we provide an alternative way to constructing a coupling of Brownian motions by space-time parallel transport. 12

13 In this section, we first define a coupling of geodesic random walks which approximate gτ- Brownian motion. Next, we introduce some estimates on geometric quantities in subsection 5.1. Those are obtained as a small modification of existing arguments in 6, 31, 36. The L-cut locus is also reviewed and studied there. We use those estimates in section 5.2 to study the behaviour of the L-distance between coupled random walks. The argument there includes a discrete analogue of the Itô formula as well as a local uniform control of error terms. Finally, we will complete the proof of Theorems 2 and 3 in section 5.3. Let us take a family of minimal L-geodesics {γ τ 1τ 2 xy τ 1 τ 1 < τ 2 τ 2, x, y M} so that a map x, τ 1 ; y, τ 2 γ τ 1τ 2 xy is measurable. The existence of such a family of minimal L-geodesics can be shown in a similar way as discussed in the proof of 21, Proposition 2.6 since the family of minimal L-geodesics with fixed endpoints is compact cf. 6, the proof of Lemma For each τ τ 1, T, take a measurable section Φ τ of gτ-orthonormal frame bundle O gτ M of M. For x, y M and τ 1, τ 2 τ 1, T with τ 1 < τ 2, let us define Φ i x, τ 1 ; y, τ 2 FM for i = 1, 2 by Φ 1 x, τ 1 ; y, τ 2 := Φ τ 1 x, Φ 2 x, τ 1 ; y, τ 2 := m τ 1τ 2 xy Φ τ 1 x, where m τ 1τ 2 xy is as given in Definition 2. Let us take a family of R d -valued i.i.d. random variables λ n n N which are uniformly distributed on a unit ball centered at origin. We denote the Riemannian exponential map with respect to gτ at x M by exp τ x. In what follows, we define a coupled geodesic random walk X ε t = X ε τ 1 t, Y ε τ 2 t with scale parameter ε > and initial condition X ε s = x 1, y 1 inductively. First we set X ε τ 1 s, Y ε τ 2 s := x 1, y 1. For simplicity of notations, we set t n := s + ε 2 n T/ τ 2. After we defined X ε t t s,tn, we extend it to X ε t t s,tn+1 by ˆλ i n+1 := d + 2Φ i X ε τ 1 t n, τ 1 t n ; Y ε τ 2 t n, τ 2 t n λ n+1, i = 1, 2, X ε τ 1 t := exp τ 1t n t tn X ε τ 1 tn ε 2 τ1ˆλ1 n+1, Y ε τ 2 t := exp τ 2t n t tn Y ε τ 2 tn ε 2 τ2ˆλ2 n+1 for t t n, t n+1. We can and we will extend the definition of Xτ ε for τ T τ 1 / τ 2, T in the same way. As in section 3, X ε τ 1 t does not move when τ 1 =. Note that the term d + 2 in the i definition of ˆλ n+1 is a normalization factor in the sense Cov d + 2λ n = Id. Let us equip path spaces Ca, b M or Ca, b M M with the uniform convergence topology induced from gt. Here the interval a, b will be chosen appropriately in each context. As shown in 15, X ε τ τ τ1 s,t and Y ε τ τ τ2 s,t converge in law to gτ-brownian motions X τ τ τ1 s,t and Y τ τ τ2 s,t on M with initial conditions X τ1 s = x 1, Y τ2 s = y 1 respectively as ε when τ 1 >. As a result, X ε is tight and hence there is a convergent subsequence of X ε. We fix such a subsequence and use the same symbol X ε ε for simplicity of notations. We denote the limit in law of X ε as ε by X t = X τ1 t, Y τ2 t. Recall that, in this paper, gτ-brownian motion means a time-inhomogeneous diffusion process associated with gτ instead of gτ /2. Remark 7. We explain the reason why our alternative construction works efficiently to avoid the obstruction arising from singularity of L. To make it clear, we begin with observing the essence of difficulties in the SDE approach we used in section 3. Recall that our argument is based on the Itô formula. Hence the non-differentiability of L at the L-cut locus causes the technical difficulty. One possible strategy is to extend the Itô formula for L-distance. Since L- cut locus is sufficiently thin, we can expect that the totality of times when our coupled particles 13

14 stay there has measure zero. In addition, as that of Riemannian cut locus, the presence of L-cut locus would work to decrease the L-distance between coupled particles. Thus one might think it possible to extend Itô formula for L-distance to the one involving a local time at the L-cut locus. If we succeed in doing so, we will obtain a differential inequality which implies the supermartingale property by neglecting this additional term since it would be non-positive. Instead of completing the above strategy, our alternative approach in this section directly provides a difference inequality without extracting the additional local time term. When the endpoint of minimal L-geodesic is in the L-cut locus, we divide it into two pieces. Then the pair of endpoints of each piece is not in the L-cut locus. As a result, we obtain the desired difference inequality of L-distance even in such a case see the proof of Lemma 4 for more details. In order to follow such a procedure, it is more suitable to work with discrete time processes. 5.1 Preliminaries on properties of L-functional Recall that we assumed the uniform lower Ricci curvature bound 2. On the basis of it, we can compare Riemannian metrics at different times. That is, for τ 1 < τ 2, gτ 1 e 2Kτ 2 τ 1 gτ 2 23 see 3, Lemma 5.3.2, for instance. Recall that ρ gτ is the distance function on M at time τ. Note that a similar comparison between ρ gτ1 and ρ gτ2 follows from 2. By neglecting the term involving γ in the definition of Lγ, the condition 2 implies inf Lx, τ 1; y, τ 2 2dK τ 3/2 2 τ 3/ x,y M 3 We also obtain the following bounds for L from 2 and 23. Let γ : τ 1, τ 2 M be a minimal L-geodesic. Then, for τ τ 1, τ 2, e 2KT 2 τ 2 τ 1 ρ g τ 1 γτ 1, γτ 2 2 3/2 dkτ2 τ 3/2 1 Lγτ 1, τ 1 ; γτ 2, τ see 6, Lemma 7.13 or 31, Proposition B.2. The same estimate holds for ρ g τ1 γτ, γτ 2 2 instead of ρ g τ1 γτ 1, γτ 2. Taking the fact that L-functional is not invariant under reparametrization of curves into account, we will introduce a local estimate on the velocity of the minimal L-geodesic γ. Lemma 3. Let τ 1, τ 2 τ 1, T and suppose that τ 2 τ 1 δ for some δ >. Then, for any compact set M M, there exist constants C 1 > depending on K, M and δ such that, for any γ : τ 1, τ 2 M with γτ 1, γτ 2 M and τ 1 τ τ 2, τ γτ 2 gτ C Proof. Though the conclusion follows by combining arguments in 6, Lemma 7.24 and 36, Proposition 2.12, we give a proof for completeness. Let o M be a reference point and take r > so large that B gt r /2 o contains M. Take K > so that sup τ R gτ K holds on Br gt o. We claim that there exists a constant C > such that Lx, τ 1 ; y, τ 2 C 27 14

15 for any x, y M. Take a constant speed gt -minimal geodesic γ : τ 1, τ 2 M joining x and y. Note that γ is contained in Br gt o. Thus, by virtue of 23, we have Lx, τ 1 ; y, τ 2 τ2 τ 1 τ 2e2KT 3 γ τ 2 gτ + R gτγ τ dτ τ 3/2 2 τ 3/2 1 τ 2 τ 1 2 d gt x, y 2 + 2K 3 2T 3/2 e 2KT 3δ 2 d gt x, y 2 + 2K 3 T 3/2. τ 3/2 2 τ 3/2 1 Thus the claim follows since x, y M. By combining the above claim with 25, we can show that there exists r 1 > r which is independent of γ such that γτ Br gt 1 o for any τ τ 1, τ 2. Take K 1 > so that Ric gτ gτ K 1 and R gτ gτ K 1 hold on Br gt 1 o for any τ τ 1, T. By a similar argument as in 6, Lemma 7.13 ii, there exists τ τ 1, τ 2 such that τ γτ 2 gτ 1 2 τ 2 τ 1 Lγτ 1, τ 1 ; γτ 2, τ 2 + 2dK 3/2 τ2 τ 3/ By virtue of 2, there exist constants c 1, C 1 > which depends on K, K 1 and T such that for all τ 1, τ 2 τ 1, τ 2 with τ 1 < τ 2, τ 2 γτ 2 2 gτ 2 c 1τ 1 γτ 1 2 gτ 1 c 1τ 2 τ 1 γτ 1 2 gτ 1 + C 1, 29 γτ 2 2 gτ 2 + C 1. 3 The first inequality in 29 can be shown similarly as 6, Lemma It is due to a differential inequality based on the L-geodesic equation 7 which provides an upper bound of τ τ γτ 2 gτ. By considering a lower bound of the same quantity instead, we obtain the second inequality 3 in a similar way. Hence the proof is completed by combining 29 and 3 with 28 and 27. Let us recall that the L-cut locus, denoted by LCut, is defined as a union of two different kinds of sets see 36; see 6, 31 also. The first one consists of x, τ 1 ; y, τ 2 such that there exists more than one minimal L-geodesics joining x, τ 1 and y, τ 2. The second consists of x, τ 1 ; y, τ 2 such that y, τ 2 is conjugate to x, τ 1 along a minimal L-geodesic with respect to L-Jacobi field. Note that L is smooth on M \ LCut see 36, Lemma 2.9 and that LCut is closed see 31; though they assumed M to be compact, an extension to the non-compact case is straightforward. 5.2 Variations of the L-distance of coupled random walks For proving Theorem 2, our first task is to show a difference inequality of Λt, X ε t in Lemma 4. We begin with introducing some notations. Set γ n := γ τ 1t n, τ 2 t n X ε and let us define a vector field tn ˆλ n+1 along γ n by ˆλ n+1 τ = τ/t n λ n+1 τ, where λ n+1 is a space-time parallel vector field along γ n with initial condition ˆλ n+1 τ 1 1t n = ˆλ n+1. Let us define random variables ζ n and Σ n as 15

16 follows: ζ n+1 := τ 2τ ˆλ n+1 τ, γ 2 t n nτ gτ, τ= τ 1 t n Σ n+1 := 1 τ τ 3/2 R t gτ γ n τ γ n τ 2 2 t n gτ n τ= τ 1 t n τ + 2 τ τ Ric gτ ˆλ n+1 τ, ˆλ 2 t n n+1 τ t n τ= τ 1 t n τ2 t n τh τ 1 t n γτ, ˆλ n+1 τ dτ Here H is as given in 13. The term ζ n+1 corresponds to the martingale part of Λt, X t and Σ n does to the one dominating the bounded variation part of Λt, X t, Ỹt in section 3. As we will see in Lemma 4 below, there is a discrete analogue of the Itô formula and the corresponding difference inequality involving ζ n and Σ n. As a result of our discretization, we are no longer able to apply Proposition 1 directly to estimate Σ n itself. In this case, we can do it to the conditional expectation of Σ n instead. Set G n := σλ 1,..., λ n. Then, since each Φ i is isometry and d + 2E λ n, e i λ n, e j = δ ij, Proposition 1 yields For M M, we define σ M E Σ n+1 G n d tn τ2 τ 1 1 2t n Λt n, X ε t n. 31 : Cs, T/ τ 2 M M, by σ M w, w := inf {t s w τ1 t / M or w τ2 t / M }. For simplicity of notations, we denote σ M X ε and σ M X by σ ε M and σ M respectively. As shown in 15, for any η >, we can take a compact set M M such that lim ε Pσ ε M T η holds cf. 17. Lemma 4. Let M M be a compact set. Then there exist a family of random variables Q ε n n N,ε> and a family of deterministic constants δε ε> with lim ε δε = satisfying Q ε n δε 32 such that n; t n<σ ε M T/ τ 2 Λt n+1, X ε t n+1 Λt n, X ε t n + εζ n+1 + ε 2 Σ n+1 + Q ε n Proof. When X ε τ 1 t n, τ 1 t n ; Y ε τ 2 t n, τ 2 t n / LCut, the inequality 33 follows from the Taylor expansion with the error term Q ε n+1 = oε2. Indeed, the first variation formula 6, Lemma 7.15 cf. 8 produces εζ n+1 and Corollary 2 together with 9 implies the bound ε 2 Σ n+1 of the second order term. To include the case X ε τ 1 t n, τ 1 t n ; Y ε τ 2 t n, τ 2 t n LCut as well as to obtain a uniform bound 32, we extend this argument. Set τ n := τ 1 + τ 2 t n /2. Then we can show X ε τ 1 t n, τ 1 t n ; γ n τ n, τ n / LCut, γ n τ n, τ n; X ε τ 2 t n, τ 2 t n / LCut since minimal L-geodesics with these pair of endpoints can be extended with keeping its minimality cf. see 6, Section 7.8 and 36. Set x n+1 = expτ n γ τ1 nτ + τ n 2 λ n+1 τ n. The triangle inequality for L yields Λt n, X ε t n = LX ε τ 1 t n, τ 1 t n ; γ n τn, τn + Lγ n τn, τn; X ε τ 2 t n, τ 2 t n, Λt n+1, X ε t n+1 L X ε τ 1 t n+1, τ 1 t n+1 ; x n+1, τn+1 + L x n+1, τn+1; X ε τ 2 t n+1, τ 2 t n

17 Hence Λt n+1, X ε t n+1 Λt n, X ε t n L X ε τ 1 t n+1, τ 1 t n+1 ; x n+1, τn+1 L X ε τ 1 t n, τ 1 t n ; γ n τn, τn + L x n+1, τn+1; X ε τ 2 t n+1, τ 2 t n+1 L γ n τn, τn; X ε τ 2 t n, τ 2 t n and the desired inequality with Q ε n = oε 2 holds by applying the Taylor expansion to each term on the right hand side of the above inequality. We turn to showing the claimed control 32 of the error term Q ε n. Take a compact set M 1 M such that every minimal L-geodesic joining x, τ 1 t and y, τ 2 t is included in M 1 if x, y M and t s, T/ τ 2. Indeed, such M 1 exists since we have the lower bound of L in 25 and L is continuous. Let us define a set A by x, y M, A := τ 1, x, τ 3, z, τ 2, y τ 1, T M 1 3 τ 2 τ 1 τ 2 τ 1 s, τ 3 = τ 1 + τ 2 /2,. Lx, τ 1 ; z, τ 3 + Lz, τ 3 ; y, τ 2 = Lx, τ 1 ; y, τ 2 Note that A is compact. Let π 1, π 2 : A τ 1, T M 1 2 be defined by π 1 τ 1, x, τ 3, z, τ 2, y := τ 1, x, τ 3, z, π 2 τ 1, x, τ 3, z, τ 2, y := τ 3, z, τ 2, y. Then π 1 A and π 2 A are compact and π i A LCut = for i = 1, 2. The second assertion comes from the fact that z, τ 3 is on a minimal L-geodesic joining x, τ 1 and y, τ 2 for x, τ 1, z, τ 3, y, τ 2 A. Recall that LCut is closed. Thus we can take relatively compact open sets G 1, G 2 τ 1, T M such that π i A G i and Ḡi LCut = for i = 1, 2. Then the Taylor expansion we discussed above can be done on G 1 or G 2 for sufficiently small ε. Recall that L is smooth outside of LCut see 6. Thus the convergence ε 2 Q n ε as ε is uniform in n and independent of X ε t n as long as t n < σm ε T/ τ 2. Since the cardinality of {n t n < σm ε T/ τ 2 } is of order at most ε 2, the assertion 32 holds. We next establish the corresponding difference inequality for Θ t X ε t Corollary 3. For that, we show the following auxiliary lemma. Lemma 5. Let M M be a compact set. Then there exist deterministic constants C 2 > depending on M such that max { ζ n, Λt n, X ε t n, Σ n } C 2 holds if t n σ ε M T/ τ 2. Proof. By the definition of ζ n, we have ζ n 2d + 2t n 1 τ1 γ n 1 τ 1 t n 1 g τ1 t n 1 + τ 2 γ n 1 τ 2 t n 1 g τ2 t n 1. Thus the asserted bound for ζ n follows from 26 and 23. Similarly, the estimate for Λt n, X ε t n follows from 25 and 23. For estimating Σ n, we deal with the integral involving H in the definition of Σ n. Note that every tensor field appeared in the definition of H is continuous. As in the proof of Lemma 4, take a compact set M 1 M such that every minimal L-geodesic joining x, τ 1 t and y, τ 2 t is included in M 1 if x, y M and t s, T/ τ 2. Since X ε t n 1 M M holds on the event {t n < σ ε M T/ τ 2 }, the upper bound 26 of τ γτ implies that H γ n τ, Zτ is uniformly bounded for any vector field Zτ along γ n of the form Zτ = τ/t n Z τ with a space-time parallel vector field Z τ satisfying Z τ gτ 1. This fact yields a required bound for the integral. For any other terms in the definition of Σ n, we can estimate them as we did for ζ n and Λt n, X ε t n. 17

18 By virtue of Lemma 5, Lemma 4 yields the following: Corollary 3. Let M M be a compact set. Then there exist a family of random variables Q ε n n N,ε> and a family of deterministic constants δε ε> with lim ε δε = satisfying Q ε n δε such that n; t n<σ ε M T/ τ 2 Θ tn+1 X ε t n+1 Θ tn X ε t n + ε2 tn τ2 τ 1 Λtn, X ε t n 2ε 2 d τ 2 τ ε For u s, T/ τ 2, let us define u ε by τ2 t n+1 τ 1 t n+1 ζ n+1 + 2ε 2 τ2 t n+1 τ 1 t n+1 Σ n+1 + Q ε n u ε := sup{s + ε 2 n n N {}, 1 + ε 2 n < u}. Set ˆσ ε M := σ ε M ε + ε 2. Note that {ˆσ ε M = t n } G n for all n N {}. We finally prepare the following moment bound of Θ t X t before entering the proof of Theorem 2. Lemma 6. There exists c 3, C 3 > such that E sup Θ t X t 2 s t T/ τ 2 < c 3 Θ s x 1, y C 3. Proof. Recall that Θ is uniformly bounded from below by 24. Take b so large that Θ t x, y + b for x, y M and t s, T/ τ 2 and set ˆΘ t x, y := Θ t x, y + b. It suffices to show E sup ˆΘt X t 2 s t T/ τ 2 4 ˆΘ s x 1, y C for some C >. Take a relatively compact open set M M and consider ˆΘ tn ˆσ ε M X ε t n ˆσ ε M. Lemma 5 ensures that the term appeared in 34 is integrable on the event t n < ˆσ M ε. Thus Corollary 3 and 31 yields that E ˆΘtm ˆσ M ε X ε t m ˆσ M ε ˆΘ tn ˆσ ε X ε M t n ˆσ M ε G n δε. By imitating the proof of the maximal inequality cf. 28, Chapter2, Exercise 1.15, we obtain P sup ˆΘ tn X ε t n r 1 δε ˆΘs x 1, y 1 + n; s t n ˆσ M ε T/ τ 2 r for r >. Then, by following the proof of the Doob inequality in 28, E sup ˆΘ tn X ε t n R 2 δε 2 4 ˆΘs x 1, y s t n ˆσ M ε T/ τ 2 holds for each R >. By 23 and the definition of X ε, there exist C M > such that ˆΘ t σ ε M X ε t R 2 ˆΘ tn X ε t n ˆσ ε M R 2 + C M ε 18

19 for t t n, t n+1. Thus 35 yields E sup ˆΘ t X ε t R 2 δε 2 4 ˆΘs x 1, y CM ε. 36 s t σm ε T/ τ 2 Let us turn to estimate the second moment of sup t ˆΘt X t. Note that we have E sup ˆΘ t X t R 2 s t T/ τ 2 E sup ˆΘ t X t R 2 ; σm > t s t T/ τ 2 + R 2 Pσ M t. 37 Since {w σ M w > t} is open and the map w sup s t T/ τ2 ˆΘ t w t R 2 is bounded and continuous on Cs, T/ τ 2 M M, 36 yields E sup ˆΘ t X t R 2 ; σm > t s t T/ τ 2 lim inf ε lim inf ε 4 ˆΘ s x 1, y 1. E sup ˆΘ t X ε t R 2 ; σm ε > t s t T/ τ 2 E sup ˆΘ t X ε t R 2 s t σm ε T/ τ 2 Thus the conclusion follows by combining the last inequality with 37 and by letting M M and R. 5.3 Proof of Theorems 2 and 3 Proof of Theorem 2. First we remark that the map x, y X ε τ 1, Y ε τ 2 is obviously measurable. Thus, we obtain the same measurability for the law of X τ1, Y τ2. The integrability of Θ t X t follows from Lemma 6. We will show the supermartingale property in the sequel. For s s 1 < < s m < t < t < T and f 1,..., f m C c M M R with f j 1, Set F w := m j=1 f jw sj for w Cs, T/ τ 2 M M. Take η > arbitrarily and choose a relatively compact open set M M so that PσM t η holds. Note that lim sup ε PσM ε t η also holds since {w σ M w t} is closed. It suffices to show that there is a constant C > which is independent of η and M such that, E Θ t σ M X t σ M Θ t σ M X t σ M holds. In fact, once we have shown 38, then Lemma 6 yields E Θ t X t Θ s X s F X F X σ C η 38 M since σ M almost surely as M M. Take f C c M M such that f 1 and f U 1, where U M M is a open set containing M M. Then, by virtue of Lemma 6 and the choice of M, E Θ t σ X M t σ Θ M t σm X t σ M F X σ M E Θ t X t Θ t X t fx t fx t F X ; σm > t + 2C 1/2 4 η, 39 19

20 where C 4 := c 3 Θ s x 1, y C 3. Since {w σ M w > t} is open, E Θ t X t Θ t X t fx t fx t F X ; σ M > t lim inf ε = lim inf ε E Θ t X ε t Θ t X ε t fxε tfx ε t F Xε ; σm ε > t E Θ t ε X ε t ε Θ t ε X ε t ε fx ε t ε fx ε t ε F X ε ; σm ε > t. 4 Here the last equality follows from the continuity of Θ and f. Then E Θ t ε X ε t ε Θ t ε X ε t ε fx ε t ε fx ε t ε F X ε ; σm ε > t E Θ t ε ˆσ M ε X ε t ε ˆσ M ε Θ t ε ˆσ ε X ε M t ε ˆσ M ε F X ε ˆσ ε M 1/2 + 2E sup Θ u X ε ufx ε u 2 PσM ε t 1/2. 41 s u T/ τ 2 Since a function w sup 1 u T/ τ2 Θ u w u fw u on Cs, T/ τ 2 M M is bounded and continuous, we have 1/2 lim sup E sup Θ u X ε ufx ε u 2 PσM ε t 1/2 C 1/2 4 η. 42 ε s u T/ τ 2 Now, with the aid of Lemma 5, the iteration of 34 together with 31 yields E Θ t ε ˆσ M ε X ε t ε ˆσ M ε Θ t ε ˆσ ε X ε M t ε ˆσ M ε F X ε ˆσ ε δε. 43 M Here δε is a constant appeared in Corollary 3. Hence we complete the proof by combining 4, 41, 42 and 43 with 39. Proof of Theorem 3. Fix 1 s < t T/ τ 2. We may assume T ϕ Θs p τ1 s, vol g τ1 t, q τ 2 s, vol g τ2 t < without loss of generality. Let π be a minimizer of T ϕ Θs p τ1 s, vol g τ1 t, q τ 2 s, vol g τ2 t, where the existence of π follows from 32, Theorem 4.1, using the lower bound 24. For each x, y M M, we take coupled Brownian motions Xτ x τ τ1 s,t and Yτ y τ τ2 s,t with initial values X x τ 1 s = x and Yτ ȳ 2 s = y as in Theorem 2. Since the law of X x, Y y is a measurable function of x, y, we can construct a coupling of two Brownian motions X τ1, Y τ2 with initial distribution π from X x τ 1, Yτ ȳ 2 x,y M as a coordinate process on Cs, T/ τ 2 M M by following a usual manner. By Theorem 2, ϕ Θ t X x τ1 t, Y ȳ τ 2 t is a supermartingale. Hence we have E ϕ Θ t X τ1 t, Y τ2 t = E ϕ Θ t X x τ1 t, Y ȳ τ 2 πdx, dy M M ϕ Θ s x, y πdx, dy M M = T ϕ Θs p τ1 s, vol g τ1 s, q τ 2 s, vol g τ2 s. Since the law of X τ1 t, Y τ2 t is a coupling of p τ 1 t, vol g τ1 t and q τ 2 t, vol g τ2 t, we have T ϕ Θt p τ1 t, vol g τ1 t, q τ 2 t, vol g τ2 t E ϕθt X τ1 t, Y τ2 t and hence the conclusion follows. Acknowledgement. We would like to thank Anton Thalmaier and an anonymous referee for very useful comments. 2

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