A coupling of Brownian motions in the L 0 -geometry
aa r X i v : . [ m a t h . P R ] A ug A COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY TAKAFUMI AMABA AND KAZUMASA KUWADA
Abstract.
Under a complete Ricci flow, we construct a couplingof two Brownian motion such that their L -distance is a super-martingale. This recovers a result of Lott [J. Lott, Optimal trans-port and Perelman’s reduced volume, Calc. Var. Partial Differ-ential Equations 36 (2009), no. 1, 49–84.] on the monotonicity of L -distance between heat distributions. Introduction
Since Perelman’s pioneering article [21], there are several attempts tostudy Ricci flow in connection with heat distributions. As one of them,J. Lott [18] provides several monotonicity formulae from the viewpointof optimal transportation by extending Topping’s approach [20, 25].His argument is based on the Eulerian calculus, which can be rigorousif everything is sufficiently regular. On the other hand, sometimes it isnot easy to verify the required regularity. For instance, we must takemuch care on it if the manifold is non-compact.Among results in [18], Lott introduced L -functional on the spaceof (space-time) curves and the associated L -distance as an analog of L -distance in [21, 25]. He proved the monotonicity of transportationcost given in terms of L -distance between heat distributions. By usingit, he gave an alternative proof of the monotonicity of F -functional in[21]. The main purpose of this paper is to prove the monotonicity of L -transportation cost by a probabilistic approach using a coupling ofBrownian motions. As an advantage of our approach, we just requiremuch weaker regularity assumptions and we can extend the result tomore general transportation costs.The organization of this paper is as follows. In the next subsection,we will state our framework and results more precisely. We also reviewLott’s result on the monotonicity of L -transportation cost there. Insubsection 1.2, we give a review of historical background and relatedknown results in more detail. All necessary calculations, formulae andproperties on L are summarized in section 2. The reader, who wants Mathematics Subject Classification(2010)
Primary 53C21; Secondary53C44, 58J65, 60J05, The first author was supported by JSPS KAKENHI GrantNumber 24 · to grasp only heuristics or a rough story of this paper, can skip section2 except for Proposition 2.7, where we give a Hessian estimate for the L -distance. The most part of section 2 (e.g., the L -cut locus, the L -exponential and so on) are analogous to ones for Riemannian distanceor L -distance.In sections 3 and 4, we will construct a coupling of g ( τ )-Brownianmotions which satisfies a requirement of our main theorem (Theo-rem 1.1 below). In section 3 we will discuss it under a strong regu-larity assumption on L . More precisely, we assume that L -cut locusis empty there (see section 2 for the definition of L -cut locus; It is aconcept analogous to the cut locus for the Riemannian distance func-tion). While it is very restrictive, we believe that the argument givenin that section will be insightful and that it helps us to understand therigorous argument given in section 4. We employ an approximation ofBrownian motions by geodesic random walks there, as in the previousresult [15] (see section 1.2 for known results as well as the reason whywe choose it).1.1. Framework and main results.
Let ( g ( t )) ≤ t ≤ T be a Ricci flowon a connected manifold M without boundary with d := dim M ≥ g ( t ) solves the following evolution equation: ∂∂t g ( t ) = − g ( t ) . (1.1)In the sequel, we always assume that ( M, g ( t )) is a complete Riemann-ian manifold for each t .For stating Lott’s result and ours, we introduce L -functional andsome notions concerning with it. Under the Ricci flow g ( t ) on d -dimensional manifold M , L -functional is given by L ( γ ) := 12 Z t ′′ t ′ (cid:8) | ˙ γ ( t ) | g ( t ) + R g ( t ) ( γ ( t )) (cid:9) d t for each piecewise smooth curve γ : [ t ′ , t ′′ ] → M , where R g ( t ) is thescalar curvature with respect to g ( t ). As a minimal value of L -functional with the fixed endpoints (in space-time), we define L -distance L . That is, for 0 ≤ t ′ < t ′′ ≤ T and m ′ , m ′′ ∈ M , L t ′ ,t ′′ ( m ′ , m ′′ ) isgiven by L t ′ ,t ′′ ( m ′ , m ′′ ) := inf γ L ( γ ) , m ′ , m ′′ ∈ M, where the infimum is taken over piecewise smooth curves γ : [ t ′ , t ′′ ] → M such that γ ( t ′ ) = m ′ and γ ( t ′′ ) = m ′′ . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 3 We denote by P ( M ) the space of Borel probability measures on M and P ∞ ( M ) the subspace of P ( M ) whose element has smooth den-sity. For 0 ≤ t ′ < t ′′ ≤ T and µ ′ , µ ′′ ∈ P ( M ), the (optimal) L -transportation cost C t ′ ,t ′′ ( µ ′ , µ ′′ ) is defined by C t ′ ,t ′′ ( µ ′ , µ ′′ ) := inf π ∈ Π( µ ′ ,µ ′′ ) Z M × M L t ′ ,t ′′ ( m ′ , m ′′ ) π (d m ′ , d m ′′ ) , where Π( µ ′ , µ ′′ ) is the set of couplings of µ ′ and µ ′′ .In this framework, Lott proved the following: Assume that M isclosed. If c ′ , c ′′ : [ t , t ] → P ∞ ( M ) are solutions to the backward heatequation d µ t d t = −△ g ( t ) µ t , (1.2)and if they satisfy some technical assumptions (see Corollary 5 in [18]for details, although we will mention it partially in the sequel), then u C t ′ + u,t ′′ + u (cid:0) c ′ ( t ′ + u ) , c ′′ ( t ′′ + u ) (cid:1) (1.3)is non-decreasing (Proposition 13 in [18]).Let us turn to state our result. Until the end of the paper, we fixtwo time intervals0 ≤ t ′ < t ′ ≤ T and 0 ≤ t ′′ < t ′′ ≤ T with t ′ < t ′′ and t ′ − t ′ = t ′′ − t ′′ =: S . We denote by ( t ′ , t ′′ ) thecoordinate on (cid:8) ( t ′ , t ′′ ) ∈ [ t ′ , t ′ ] × [ t ′′ , t ′′ ] : t ′ < t ′′ (cid:9) . Since it looks awkward to work with backward heat equation (1.2),we shall reverse the time by setting τ ′ = τ ′ ( s ) := t ′ − s and τ ′′ = τ ′′ ( s ) := t ′′ − s for 0 ≤ s ≤ S . By g ( τ ′ ) -Brownian motion (resp. g ( τ ′′ ) -Brownianmotion) , we mean a time-inhomogeneous diffusion process on M as-sociated with △ g ( τ ′ ( s )) (resp. △ g ( τ ′′ ( s )) ), where s stands for the time-parameter of the process. Theorem 1.1.
Assume that ( g ( t )) ≤ t ≤ T satisfies inf ( t,m ) ∈ [0 ,T ] × M inf V ∈ T m M k V k g ( t ) =1 Ric g ( t ) ( V, V ) > −∞ . (1.4) Then for each ( m ′ , m ′′ ) ∈ M × M , there exists a coupling of g ( τ ′ ) -Brownian motion X = ( X s ) ≤ s ≤ S starting from m ′ and g ( τ ′′ ) -Brownianmotion Y = ( Y s ) ≤ s ≤ S starting from m ′′ such that s L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is a supermartingale and the map ( m ′ , m ′′ ) the law of ( X, Y ) with ( X , Y ) = ( m ′ , m ′′ ) TAKAFUMI AMABA AND KAZUMASA KUWADA is measurable.
Theorem 1.1 provides us a probabilistic interpretation of the mono-tonicity of (1.3). That is, roughly speaking, we can show the mono-tonicity by taking an expectation of L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ). Actually, wecan say more: Let ϕ : R → R be concave and non-decreasing. Wedefine a new transportation cost C t ′ ,t ′′ ,ϕ ( µ ′ , µ ′′ ) by C t ′ ,t ′′ ,ϕ ( µ ′ , µ ′′ ) := inf π ∈ Π( µ ′ ,µ ′′ ) Z M × M ϕ (cid:0) L t ′ ,t ′′ ( m ′ , m ′′ ) (cid:1) π (d m ′ , d m ′′ ) . Corollary 1.2.
Assume that our Ricci flow satisfies the condition (1.4) . Then for any two families c ( t ′ ) and c ( t ′′ ) of probability mea-sures satisfying (1.2) , C t ′ + s,t ′′ + s ,ϕ (cid:0) c ′ ( t ′ + s ) , c ′′ ( t ′′ + s ) (cid:1) is nondecreasingin s , Note that Lott’s result can be regarded as a special case of Corol-lary 1.2, that is, the case ϕ ( x ) = x and M is compact. Moreover, inorder to make Otto’s calculus rigorous, Lott further assumed that thecurves c ′ , c ′′ and the “ E -minimizing geodesics” (see (4.9) in [18] for thedefinition of E ) which interpolate c ′ ( t ′ + s ) and c ′′ ( t ′′ + s ) lie in P ∞ ( M ).In [18], it is claimed that the last extra condition can be relaxed bygiving an alternative proof which is analogous to Topping’s approachin [25]. In this paper, we give a proof based on the theory of stochasticcalculus, and we do not only relax the extra regularity assumption butweaken the compactness assumption on M to the curvature condition(1.4). Recall that, under a well-known sufficient condition to the uniqueexistence of a Ricci flow in [4, 23], the condition (1.4) is automaticallysatisfied. As a matter of fact, stochastic calculus is already used in[15] to extend Topping’s result [25] (see below for more details). Thusour result provides an additional evidence that stochastic calculus isan efficient tool to study this sort of problems.1.2. Historical background and related results.
Recall that theRicci flow is a solution to the evolution equation (1.1). This equationwas introduced by Hamilton in [9] and used to find an Einstein metric(i.e., a metric g such that Ric( g ) = const .g ) by deforming any givenRiemannian metric g with positive Ricci curvature on a compact 3-manifold.Inspired by quantum field theory, such as nonlinear σ -models, Perel-man [21] interprets the Ricci flow as a gradient flow ; At least formally,the Ricci flow can be regarded as the gradient flow of the so-calledPerelman’s F -functional . This interpretation naturally leads to the monotonicity formula for F : The functional F is nondecreasing alongthe Ricci flow. Additionally, the so-called W -functional is also shown tobe non-decreasing along the Ricci flow. As is well-known, these mono-tonicity formulae are effectively used in the resolution of Thurston’sgeometrization conjecture by Perelman. COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 5 Recently, alternative approaches to those formulae have been initi-ated on the basis of optimal transportation. For the monotonicity of W ,Topping [25] gave an alternative proof by using the L -transportationcost via so-called Lagrangian calculus. More precisely, he consider theoptimal transportation cost whose cost function is given by a renor-malization of Perelman’s L -distance. He proved the monotonicity ofthis transportation cost between (time-rescaled) heat distributions andderived the monotonicity of W -functional by taking a sort of time-derivative of the optimal cost. For studying the monotonicity of F ,Lott [18] showed the monotonicity of (1.3) as explained in section 1.1.Then he recovered the monotonicity of F -functional (Corollary 6 in[18]) again by taking a sort of time-derivative. As mentioned, Lott’sargument is based on so-called Eulerian calculus (see e.g. [26] for acomparison with Lagrangian calculus).Such monotonicity formulae for optimal transportation costs be-tween heat distributions are also studied in a slightly different context.For instance, on a complete Riemannian manifold with a fixed metric,the same sort of monotonicity of L p -Wasserstein distance is equivalentto non-negative Ricci curvature or an L q -gradient estimate for the heatsemigroup (see [22] and references therein). For time-dependent met-rics, the same sort of monotonicity of L -Wasserstein distance is shownto be equivalent to the property that the metric evolves as a super Ricciflow by McCann and Topping [20]. On one hand, the latter result is anatural extension of the former one since the latter recovers the formerwhen the metric does not depend on time. On the other hand, thisresult can be regarded as a primitive form of the results [18, 25] (Thisobservation is also addressed in [18, 25] themselves). These former re-sults indicate that monotonicity formulae in the optimal transportationshould be connected with the geometry of the space in a deeper wayand that more studies are expected in this direction.From its definition, optimal transportation cost is strongly relatedwith the notion of coupling of random variables or stochastic processes.Thus it is natural to consider the above-mentioned problem by using acoupling method of stochastic processes. Even only in stochastic dif-ferential geometry, there are several results in coupling method. Tra-ditionally, they paid much attention to estimating the time that twoparticle meets, while it does not match with our present purpose (see[7, 11, 17]; see [12, 27] also). By using a similar idea as ones in thosestudies, we can construct a coupling by parallel transport on a com-plete Riemannian manifold with a lower Ricci curvature bound andit is tightly connected with the monotonicity of Wasserstein distances(see [22] and references therein). Extensions of those kind of couplingto the time-dependent metric case are achieved by Kuwada [16] andby Arnaudon, Coulibaly and Thalmaier [1]. A typical example of thetime-dependent metric is backward (super) Ricci flow. To construct a TAKAFUMI AMABA AND KAZUMASA KUWADA coupling, the former used an approximation by coupled geodesic ran-dom walks and the latter construct a one-parameter family of coupledparticles which consists of a string and moves continuously as timeevolves. As a result, they recover the monotonicity formula in [20] andextend it to non-compact spaces. Topping’s monotonicity formula isalso proved and extended by Kuwada and Philipowski [15] and dis-cussed later by Cheng [5]. The former used the same method as in [16]and the latter uses an argument studied in [27].In all those couplings, we are interested in the behavior of distance-like functions (e.g. distance itself or L -distance) between a couplingof diffusion processes ( X s , Y s ). In those cases, the main technical diffi-culty arises at singular points of the functions (e.g. cut locus). Roughlyspeaking, there are two obstructions: Firstly, the construction of thecoupling itself depends on a regularity of the function. Secondly, wecan not apply Itˆo’s formula directly when the coupled process lies onthe singular points. Thus we require some indirect arguments as men-tioned above to overcome these obstructions. In this paper, we followthe argument used in [15, 16]. More precisely, we consider a couplingof geodesic random walks ( X εs , Y εs ). The construction of it requiresless regularity and this fact works well to avoid the first obstruction.Then, instead of Itˆo’s formula for L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ), we can employ a “difference” (rather than differential) inequality for L τ ′ ( s ) ,τ ′′ ( s )0 ( X εs , Y εs )at each approximation step ε (see Proposition 4.2), up to an well-controlled error. Along this idea, we can avoid the second obstruction.Since geodesic random walks converge to (time-inhomogeneous) Brow-nian motions, we can obtain an estimate for a coupling of Brownianmotions as the limit. For those who are interested in other approaches,it is worth mentioning that a comparison with other approaches is dis-cussed in [15, 16].2. L -geometry along Ricci flow In the rest of the paper, we always assume (1.4).2.1.
Differential calculus of L and L . The main aim in this sub-section is to give an estimate of (a contraction of) the Hessian for L t ′ ,t ′′ ,which we will use in the subsequent sections. Let γ : [ t ′ , t ′′ ] → M bea piecewise smooth curve. For each variation of γ with a variationalvector field V , we denote by ( δ V L )( γ ) and ( δ V δ V L )( γ ) the first andsecond variation of L . We omit all proofs in this subsection except forProposition 2.7, because all proofs are routine (see e.g., [13, section 17,18 and 19]). Proposition 2.1.
For any smooth variation (not being necessarilyproper) of a smooth curve γ : [ t ′ , t ′′ ] → M with a variational vector COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 7 field V , we have ( δ V L )( γ ) = h V ( t ) , ˙ γ ( t ) i g ( t ) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + 12 Z t ′′ t ′ h G t ( γ ) , V ( t ) i g ( t ) d t where G t ( γ ) := ∇ g ( t ) R g ( t ) − ∇ g ( t )˙ γ ( t ) ˙ γ ( t ) + 4Ric g ( t ) ( ˙ γ ( t ) , · ) . In particular, ( δ V L )( γ ) is independent of the choice of a variation which realizes thevariational vector field V as its infinitesimal variation. Hence we call (2.1) ∇ g ( t )˙ γ ( t ) ˙ γ ( t ) − ∇ g ( t ) R g ( t ) − g ( t ) ( ˙ γ ( t ) , · ) = 0 the L -geodesic equation, where the Ric g ( t ) ( ˙ γ ( t ) , · ) is naturally under-stood as a (1 , -tensor by the metric g ( t ) . We call any solution of L -geodesic equation an L -geodesic. Proposition 2.2.
Assume that L is smooth around ( t ′ , m ′ ; t ′′ , m ′′ ) andthat there exists a unique L -minimizing curve γ joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ) . Then we have ∂L t ′ ,t ′′ ∂t ′ ( m ′ , m ′′ ) = − n | ˙ γ ( t ′ ) | g ( t ′ ) + R g ( t ′ ) ( m ′ ) o − ∇ g ( t ′ )˙ γ ( t ′ ) L t ′ ,t ′′ ( · , m ′′ ) ,∂L t ′ ,t ′′ ∂t ′′ ( m ′ , m ′′ ) = 12 n | ˙ γ ( t ′′ ) | g ( t ′′ ) + R g ( t ′′ ) ( m ′′ ) o − ∇ g ( t ′′ )˙ γ ( t ′′ ) L t ′ ,t ′′ ( m ′ , · ) . Proposition 2.3.
Under the assumption in Proposition 2.2, we have ∇ g ( t ′ ) m ′ L t ′ ,t ′′ ( · , m ′′ ) = − ˙ γ ( t ′ ) and ∇ g ( t ′′ ) m ′′ L t ′ ,t ′′ ( m ′ , · ) = ˙ γ ( t ′′ ) . In particular, by combining with Proposition 2.2, ∂L t ′ ,t ′′ ∂t ′ ( m ′ , m ′′ ) = 12 n | ˙ γ ( t ′ ) | g ( t ′ ) − R g ( t ′ ) ( m ′ ) o ,∂L t ′ ,t ′′ ∂t ′′ ( m ′ , m ′′ ) = − n | ˙ γ ( t ′′ ) | g ( t ′′ ) − R g ( t ′′ ) ( m ′′ ) o . We denote the curvature tensor with respect to g ( t ) by R g ( t ) . It ap-pears in the following second variation formula for L . For a piecewisesmooth curve γ : [ t ′ , t ′′ ] → M and two vector fields V , W along γ , wedefine the L -index form L I γ ( V, W ) as follows: L I γ ( V, W ) := Z t ′′ t ′ n h∇ g ( t )˙ γ ( t ) V ( t ) , ∇ g ( t )˙ γ ( t ) W ( t ) i g ( t ) + hR g ( t ) ( V ( t ) , ˙ γ ( t )) W ( t ) , ˙ γ ( t ) i + 12 Hess R g ( t ) ( V ( t ) , W ( t )) + ( ∇ g ( t ) V Ric g ( t ) )( ˙ γ ( t ) , W ( t ))+ ( ∇ g ( t ) W Ric g ( t ) )( ˙ γ ( t ) , V ( t )) − ( ∇ g ( t )˙ γ ( t ) Ric g ( t ) ( V ( t ) , W ( t )) o d t. By definition, L I γ ( V, W ) is symmetric in V and W . TAKAFUMI AMABA AND KAZUMASA KUWADA
Proposition 2.4.
For any smooth variation (not being necessarilyproper) of an L -geodesic γ : [ t ′ , t ′′ ] → M with a variational vectorfield V , we have ( δ V δ V L )( γ ) = (cid:10) ∇ g ( t ) V ( t ) V ( t ) , ˙ γ ( t ) (cid:11) g ( t ) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + L I γ ( V, V ) . (2.2) Remark 2.1.
In (2.2), the second term on the right hand side is inde-pendent of the choice of a variation of γ which realizes the variationalvector field V as its infinitesimal variation. On the other hand, since γ is L -geodesic, the first term can be written as follows: (cid:10) ∇ g ( t ) V ( t ) V ( t ) , ˙ γ ( t ) (cid:11) g ( t ) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ = ( δ ∇ V V L )( γ ) . The next formula is derived from Proposition 2.4.
Proposition 2.5.
Keeping the notations in Proposition 2.4, we havean alternative form of the second variational formula: ( δ V δ V L )( γ )= (cid:10) ∇ g ( t ) V ( t ) V ( t ) , ˙ γ ( t ) (cid:11)(cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + Ric g ( t ) ( V ( t ) , V ( t )) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + 12 Z t ′′ t ′ n Hess g ( t ) R g ( t ) ( V ( t ) , V ( t )) + 2 D R g ( t ) ( V ( t ) , ˙ γ ( t )) V ( t ) , ˙ γ ( t ) E g ( t ) − g ( t ) d t ( V ( t ) , V ( t )) + 4 h(cid:16) ∇ g ( t ) V ( t ) Ric g ( t ) (cid:17) ( V ( t ) , ˙ γ ( t )) − (cid:16) ∇ g ( t )˙ γ ( t ) Ric g ( t ) (cid:17) ( V ( t ) , V ( t )) i + 2 (cid:12)(cid:12)(cid:12) ∇ g ( t )˙ γ ( t ) V ( t ) − Ric g ( t ) ( V ( t ) , · ) (cid:12)(cid:12)(cid:12) g ( t ) − | Ric g ( t ) ( V ( t ) , · ) | g ( t ) o d t. To deduce an estimate of (a contraction of) the Hessian for L t ′ ,t ′′ , weneed a testing vector field. For this, we introduce the notion of space-time parallel transport. This notion will be used also to construct acoupling of two Brownian motions in subsequent sections. Definition 2.6 (Space-time parallel transport along an L -minimizingcurve) . Let m ′ , m ′′ ∈ M and t ′ < t ′′ . Let γ be a L -minimizing curvejoining ( t ′ , m ′ ) and ( t ′′ , m ′′ ). We define the space-time parallel transport // t ′ ,t ′′ m ′ ,m ′′ : T m ′ M → T m ′′ M along γ as // t ′ ,t ′′ m ′ ,m ′′ ( v ) := V ( t ′′ )by solving the linear differential equation (cid:26) ∇ g ( t )˙ γ ( t ) V ( t ) = Ric g ( t ) ( V ( t ) , · ) , t ′ ≤ t ≤ t ′′ ,V ( t ′ ) = v. One can check easily that // t ′ ,t ′′ m ′ ,m ′′ gives a linear isometry from ( T m ′ M, g ( t ′ ))to ( T m ′′ M, g ( t ′′ )). Note that the space-time parallel transport can bedefined as an isometry for more general time-dependent metrics; see[15, Remark 5]. COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 9 The main result in this subsection is the following.
Proposition 2.7.
Let u ′ , · · · , u ′ d be an orthonormal basis of ( T m ′ M, g ( t ′ )) .Under the assumption in Proposition 2.2, Then it holds that d X i =1 (cid:2) Hess g ( t ′ ) ⊕ g ( t ′′ ) L t ′ ,t ′′ (cid:3)(cid:0) u ′ i ⊕ // t ′ ,t ′′ m ′ ,m ′′ u ′ i , u ′ i ⊕ // t ′ ,t ′′ m ′ ,m ′′ u ′ i (cid:1) ≤ n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o ( m ′ , m ′′ ) . For the proof, we gather formulae for geometric quantities along theRicci flow. For the proof, see [24, equation (2.1.9) and subsection 2.5].
Proposition 2.8.
Along the Ricci flow d g d t ( t ) = − g ( t ) , one has (i) ∂R g ( t ) ∂t = ∆ g ( t ) R g ( t ) + 2 | Ric g ( t ) | g ( t ) , (ii) tr dRic g ( t ) d t = ∆ g ( t ) R g ( t ) , (iii) contracted Bianchi identity: tr (cid:16) ∇ Ric g ( t ) (cid:17) = 12 ∇ g ( t ) R g ( t ) . Proof of Proposition 2.7.
Let γ be an L -minimizing curve from ( t ′ , m ′ )to ( t ′′ , m ′′ ). By Proposition 2.3, Proposition 2.8 and (2.1), we see n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o ( m ′ , m ′′ )= 12 Z t ′′ t ′ n ∆ g ( t ) R g ( t ) + 2 | Ric g ( t ) | g ( t ) − g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) o d t. Next we compute and give an estimate for (a contraction of) theHessian. For each i = 1 , , · · · , d , we define a system of vector fields( A i ) di =1 along γ by A i ( t ) := // t ′ ,tm ′ ,γ ( t ) u ′ i for t ′ ≤ t ≤ t ′′ and we take a variation f i : ( − ε , ε ) × [ t ′ , t ′′ ] → M of γ ( ε > f i (0 , · ) = γ ,(b) f i has A i as its variational field: A i (0 , t ) = A i ( t ) t ′ ≤ t ≤ t ′′ where A i ( ε, t ) := d f i d ε ( ε, t ) is the transversal vector field, and(c) two transversal curves f i ( · , t ′ ), f i ( · , t ′′ ) : ( − ε , ε ) → M are g ( t ′ )-geodesic and g ( t ′′ )-geodesic respectively at ε = 0: ∇ g ( t ′ ) A i ( t ′ ) A i ( · , t ′ ) = 0 and ∇ g ( t ′′ ) A i ( t ′′ ) A i ( · , t ′′ ) = 0 . We further set ℓ i ( ε ) := L t ′ ,t ′′ (cid:0) f i ( ε, t ′ ) , f i ( ε, t ′′ ) (cid:1) ≤ L ( f i ( ε, · )) =: b ℓ i ( ε ) . It is easy to see that ℓ ′′ i (0) ≤ b ℓ i ′′ (0). Since ∇ g ( t ′ ) ⊕ g ( t ′′ ) A i ( t ′ ) ⊕ A i ( t ′′ ) A i ( · , t ′ ) ⊕ A i ( · , t ′′ ) = (cid:8) ∇ g ( t ′ ) A i ( t ′ ) A i ( · , t ′ ) (cid:9) ⊕ (cid:8) ∇ g ( t ′′ ) A i ( t ′′ ) A i ( · , t ′′ ) (cid:9) = 0 , we can compute the Hessian as (cid:2) Hess g ( t ′ ) ⊕ g ( t ′′ ) L t ′ ,t ′′ (cid:3)(cid:0) u ′ i ⊕ // t ′ ,t ′′ m ′ ,m ′′ u ′ i , u ′ i ⊕ // t ′ ,t ′′ m ′ ,m ′′ u ′ i (cid:1) = (cid:2) Hess g ( t ′ ) ⊕ g ( t ′′ ) L t ′ ,t ′′ (cid:3)(cid:0) A i ( t ′ ) ⊕ A i ( t ′′ ) , A i ( t ′ ) ⊕ A i ( t ′′ ) (cid:1) = A i ( t ′ ) ⊕ A i ( t ′′ ) (cid:8)(cid:0) A i ( · , t ′ ) ⊕ A i ( · , t ′′ ) (cid:1) L t ′ ,t ′′ (cid:9) = d d ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 L t ′ ,t ′′ (cid:0) f i ( ε, t ′ ) , f i ( ε, t ′′ ) (cid:1) = ℓ ′′ i (0) ≤ b ℓ i ′′ (0) = ( δ A i δ A i L )( γ ) . By the second variational formula (Proposition 2.5), we have( δ A i δ A i L )( γ )= Ric g ( t ) ( A i ( t ) , A i ( t )) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + 12 Z t ′′ t ′ n Hess g ( t ) R g ( t ) ( A i ( t ) , A i ( t ))+ 2 D R g ( t ) ( A i ( t ) , ˙ γ ( t )) A i ( t ) , ˙ γ ( t ) E g ( τ ) − g ( t ) d t ( A i ( t ) , A i ( t ))+ 4 h(cid:16) ∇ g ( t ) A i ( t ) Ric g ( t ) (cid:17) ( A i ( t ) , ˙ γ ( t )) − (cid:16) ∇ g ( t )˙ γ ( t ) Ric g ( t ) (cid:17) ( A i ( t ) , A i ( t )) i + 2 (cid:12)(cid:12)(cid:12) ∇ g ( t )˙ γ ( t ) A i ( t ) − Ric g ( t ) ( A i ( t ) , · ) (cid:12)(cid:12)(cid:12) g ( t ) − | Ric g ( t ) ( A i ( t ) , · ) | g ( t ) o d t. COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 11 Hence, by taking the sum over i = 1 , , . . . , d with formulae in Propo-sition 2.8, we have d X i =1 (cid:2) Hess g ( t ′ ) ⊕ g ( t ′′ ) L t ′ ,t ′′ (cid:3)(cid:0) u ′ i ⊕ // t ′ ,t ′′ m ′ ,m ′′ u ′ i , u ′ i ⊕ // t ′ ,t ′′ m ′ ,m ′′ u ′ i (cid:1) ≤ R g ( t ) ( γ ( t )) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + 12 Z t ′′ t ′ n ∆ g ( t ) R g ( t ) − g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) − g ( t ) R g ( t ) + 4 h ∇ g ( t )˙ γ ( t ) R g ( t ) − ∇ g ( t )˙ γ ( t ) R g ( t ) i − | Ric g ( t ) | g ( t ) o d t = 12 Z t ′′ t ′ n g ( t ) R g ( t ) + 4 | Ric g ( t ) | g ( t ) + 2 ∇ g ( t )˙ γ ( t ) R g ( t ) o d τ + 12 Z t ′′ t ′ n − ∆ g ( t ) R g ( t ) − g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) − ∇ g ( t )˙ γ ( t ) R g ( t ) − | Ric g ( t ) | g ( t ) o d t = 12 Z t ′′ t ′ n ∆ g ( t ) R g ( t ) + 2 | Ric g ( t ) | g ( t ) − g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) o d t, which is equal to n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o ( m ′ , m ′′ ). (cid:3) Some estimates on relatively compact open subsets.
Webegin with estimates which hold globally under the curvature assump-tion (1.4). Let K − > − K − g ( t ) ≤ Ric g ( t ) for all t ∈ [0 , T ]. Recall that dim M = d and our Ricci flow is de-fined on [0 , T ]. Given any metric g , we denote by ρ g the correspondingRiemannian distance. Proposition 2.9. (i) Comparison of metric g ( t ) between two different times: g ( t ) ≤ e K − ( t − s ) g ( s ) for ≤ s ≤ t ≤ T . (ii) Comparison of distance ρ g ( t ) between two different times: ρ g ( t ) ( x, y ) ≤ e K − ( t − s ) ρ g ( s ) ( x, y ) for any x , y ∈ M and ≤ s ≤ t ≤ T . (iii) Lower bound for L : For ≤ t ′ < t ′′ ≤ T , m ′ , m ′′ ∈ M and a piecewise C curve γ : [ t ′ , t ′′ ] → M with γ ( t ′ ) = m ′ and γ ( t ′′ ) = m ′′ , L ( γ ) ≥
12 e − K − ( t ′′ − t ′ ) Z t ′′ t ′ | ˙ γ ( t ) | g ( t ′′ ) d t − dK − t ′′ − t ′ ) . In particular, L t ′ ,t ′′ ( m ′ , m ′′ ) ≥
12 e − K − ( t ′′ − t ′ ) ρ g ( t ′′ ) ( m ′ , m ′′ ) t ′′ − t ′ − dK − t ′′ − t ′ ) , inf ( t ′ ,m ′ ; t ′′ ,m ′′ ) ∈ ([0 ,T ] × M ) t ′ 12 e − K − ( t ′′ − t ′ ) Z t ′′ t ′ | ˙ γ ( t ) | g ( t ′′ ) d t + dK − ≥ 12 e − K − ( t ′′ − t ′ ) ρ g ( t ′′ ) ( m ′ , m ′′ ) t ′′ − t ′ + dK − . Thus the conclusion holds since γ is arbitrary. (cid:3) Proposition 2.10. Let m ′ , m ′′ ∈ M , ≤ t ′ < t ′′ ≤ T and let γ :[ t ′ , t ′′ ] → M be an L -geodesic joining ( t ′ , m ′ ) to ( t ′′ , m ′′ ) . For eachcurve η : [ t ′ , t ′′ ] → M , put K + ( η ) := inf n K > g ( t ) ≤ Kg ( t ) along η ( t ) o ,C ( η ) := sup t ∈ [ t ′ ,t ′′ ] |∇ g ( t ) R g ( t ) ( η ( t )) | g ( t ) . Then we have the following: (i) Upper bound for L : For each g ( t ′ ) -geodesic c : [ t ′ , t ′′ ] → M with c ( t ′ ) = m ′ and c ( t ′′ ) = m ′′ , L t ′ ,t ′′ ( m ′ , m ′′ ) ≤ ρ g ( t ′ ) ( m ′ , m ′′ ) t ′′ − t ′ ) e K − ( t ′′ − t ′ ) − K − + dK + ( c )( t ′′ − t ′ )2 . (ii) Bound of dd t | ˙ γ ( t ) | g ( t ) : For each t ∈ [ t ′ , t ′′ ] , we have − (cid:16) K − + 12 (cid:17) | ˙ γ ( t ) | g ( t ) − C ( γ )2 ≤ dd t | ˙ γ ( t ) | g ( t ) ≤ (cid:16) K + ( γ )+ 12 (cid:17) | ˙ γ ( t ) | g ( t ) + C ( γ )2 . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 13 (iii) Comparison of | ˙ γ ( t ) | g ( t ) between different times: There are con-stants c i > i = 1 , , , such that, for t ′ ≤ u ′ ≤ u ′′ ≤ t ′′ , | ˙ γ ( u ′′ ) | g ( u ′′ ) ≤ c | ˙ γ ( u ′ ) | g ( u ′ ) + c , | ˙ γ ( u ′ ) | g ( u ′ ) ≤ c | ˙ γ ( u ′′ ) | g ( u ′′ ) + c where the constants depend only on T , K − , K + ( γ ) and C ( γ ) . (iv) Bounding the speed of L -minimizing curve at some time by L : If the curve γ is L -minimizing, there is t ∗ ∈ ( t ′ , t ′′ ) suchthat | ˙ γ ( t ∗ ) | g ( t ∗ ) ≤ L t ′ ,t ′′ ( m ′ , m ′′ ) t ′′ − t ′ + dK − . Proof. (i) The proof is similar to Proposition 2.9(iii). By Proposition2.9(i), we see that L t ′ ,t ′′ ( m ′ , m ′′ ) ≤ L ( c ) ≤ Z t ′′ t ′ n e K − ( t − t ′ ) | ˙ c ( t ) | g ( t ′ ) + dK + ( c ) o d t. Since | ˙ c ( t ) | g ( t ′ ) ≡ ρ g ( t ′ ) ( m ′ , m ′′ ) t ′′ − t ′ , the claim holds.(ii) Using the L -geodesic equation,dd t | ˙ γ ( t ) | g ( t ) = − g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) + 2 (cid:10) ∇ g ( t )˙ γ ( t ) ˙ γ ( t ) , ˙ γ ( t ) (cid:11) g ( t ) = 2Ric g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) + (cid:10) ∇ g ( t ) R g ( t ) , ˙ γ ( t ) (cid:11) g ( t ) ≤ g ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) + |∇ g ( t ) R g ( t ) | g ( t ) + | ˙ γ ( t ) | g ( t ) ≤ (cid:16) K + ( γ ) + 12 (cid:17) | ˙ γ ( t ) | g ( t ) + C ( γ )2 . The lower bound dd t | ˙ γ ( t ) | g ( t ) ≥ − (cid:16) K − + 12 (cid:17) | ˙ γ ( t ) | g ( t ) − C ( γ )2 can beobtained similarly.(iii) By (ii), Gronwall’s lemma implies | ˙ γ ( u ′′ ) | g ( u ′′ ) ≤ e (2 K + ( γ )+ )( u ′′ − u ′ ) | ˙ γ ( u ′ ) | g ( u ′ ) + C ( γ )4 K + ( γ ) + 1 (cid:16) e (2 K + ( γ )+ )( u ′′ − u ′ ) − (cid:17) . The other is obtained similarly.(iv) By the mean value theorem, we can take t ∗ ∈ ( t ′ , t ′′ ) such that | ˙ γ ( t ∗ ) | g ( t ∗ ) = 1 t ′′ − t ′ Z t ′′ t ′ | ˙ γ ( t ) | g ( t ) d t. Since γ is L -minimizing, the right hand side is dominated by L t ′ ,t ′′ ( m ′ , m ′′ ) t ′′ − t ′ − t ′′ − t ′ ) Z t ′′ t ′ R g ( t ) ( γ ( t ))d t ≤ L t ′ ,t ′′ ( m ′ , m ′′ ) t ′′ − t ′ + dK − . (cid:3) The following is a starting point of local estimates in this subsection. Lemma 2.11. For each δ > and a relatively compact open subset M ⊂ M , there exists a relatively compact open subset B ⊃ M suchthat, for each m ′ , m ′′ ∈ M and t, t ′ , t ′′ ∈ [0 , T ] with t ′′ − t ′ ≥ δ , all L -minimizing curves joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ) and all g ( t ) -length-minimizing curves joining m ′ and m ′′ are contained in B .Proof. Since M is relatively compact, there is a compact B with M ⊂ B ⊂ M which contains all g (0)-geodesics joining any pair of points in M . Let us define K + = K + ( B ) and R := R ( B ) by K + ( B ) := inf (cid:8) K > g ( t ) ≤ Kg ( t ) on B (cid:9) ,R ( B ) := sup m ′ ,m ′′ ∈ M (cid:26) ρ g (0) ( m ′ , m ′′ ) δ e T − K − + dK + ( B ) T (cid:27) . Then, by Proposition 2.10(i) and Proposition 2.9(ii), L t ′ ,t ′′ ( m ′ , m ′′ ) ≤ R holds for all 0 ≤ t ′ < t ′′ ≤ T with t ′′ − t ′ ≥ δ and m ′ , m ′′ ∈ M . Let R = R ( B ) be defined as follows: R ( B ) := 2 T e K − T (cid:18) R + dK − T (cid:19) . Take a relatively compact open B with B ⊂ B ⊂ M such that ρ g ( T ) ( M , B c ) > p R / 2. Then, for any curve γ : [ t ′ , t ′′ ] → M with γ ( t ′ ) , γ ( t ′′ ) ∈ M and γ ([ t ′ , t ′′ ]) ∩ B c = ∅ , Proposition 2.9(iii) and Propo-sition 2.9(i) yields L ( γ ) > R . Thus B enjoys the claimed propertyon L -minimizing curves. By a similar argument, we can prove thecorresponding property for g ( t )-geodesics by using Proposition 2.9(ii).Thus, the assertion holds by enlarging B if necessary. (cid:3) Remark 2.2. By Proposition 2.10 and Lemma 2.11, we see that foreach bounded open set M ⊂ M and δ > 0, there is positive constants K + = K + ( M , δ ) > C = C ( M , δ ) > g ≤ K + g, and |∇ g ( t ) R g ( t ) ( m ) | g ( t ) < C on [0 , T ] × M ,and further, for each m ′ , m ′′ ∈ M , t ′′ − t ′ ≥ δ and each L -minimizingcurve γ joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ), we have(2.3) L t ′ ,t ′′ ( m ′ , m ′′ ) ≤ ρ g ( t ′ ) ( m ′ , m ′′ ) t ′′ − t ′ ) e K − t ′′ − e K − t ′ K − + dK + ( t ′′ − t ′ )2and(2.4) − (cid:16) K − + 12 (cid:17) | ˙ γ ( t ) | g ( t ) − C ≤ dd t | ˙ γ ( t ) | g ( t ) ≤ (cid:16) K + + 12 (cid:17) | ˙ γ ( t ) | g ( t ) + C . Lemma 2.12. ( t ′ , m ′ ; t ′′ , m ′′ ) L t ′ ,t ′′ ( m ′ , m ′′ ) is continuous on { ( t ′ , m ′ ; t ′′ , m ′′ ) | ≤ t ′ < t ′′ ≤ T, m ′ , m ′′ ∈ M } . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 15 Proof. Let 0 ≤ t ′ < t ′′ ≤ T and m ′ , m ′′ ∈ M and take ε ∈ (0 , δ > t ′′ − t ′ ≥ δ . Let U ′ and U ′′ be g (0)-metric balls ofradius δ centered at m ′ and m ′′ respectively. Let B be as in Lemma2.11 for M = U ′ ∪ U ′′ and δ = δ . Take δ ′ ∈ (0 , δ / 4) so that it satisfiesthe following: • (1 − ε ) g ( s ) ≤ g ( t ) ≤ (1 + ε ) g ( s ) and | R g ( s ) − R g ( t ) | < ε on B foreach s, t ∈ [0 , T ] with | s − t | < δ ′ , • δ ′ / ( δ − δ ′ ) < ε ,Let K + and C be as in Remark 2.2, corresponding to M and δ ′ . Take δ ∈ (0 , δ ′ ) so that dK + δ / < ε . Take δ > δ ( e K − T − e K − (2 T − δ ) )4 δ K − < ε holds and take δ = δ ∧ δ . Let m ′ , m ′′ ∈ M with ρ g (0) ( m ′ , m ′ ) ∨ ρ g (0) ( m ′′ , m ′′ ) < δ and t ′ , t ′′ ∈ [0 , T ] with | t ′ − t ′ | ∨ | t ′′ − t ′′ | < δ . Takea curve γ : [ t ′ , t ′′ ] → M from ( t ′ , m ′ ) to ( t ′′ , m ′′ ) such that L ( γ ) ≤ L t ′ ,t ′′ ( m ′ , m ′′ ) + ε . In addition, we take a curve γ from ( t ′ , m ′ ) to( t ′ + δ , m ′ ), and a curve γ from ( t ′′ − δ , m ′′ ) to ( t ′′ , m ′′ ). Let α :[ t ′ + δ , t ′′ − δ ] → [ t ′ , t ′′ ] be a (unique) affine increasing surjection. Wedefine ˜ γ : [ t ′ , t ′′ ] → M as follows:˜ γ ( t ) := γ ( t ) ( t ∈ [ t ′ , t ′ + δ )) ,γ ( α ( t )) ( t ∈ [ t ′ + δ , t ′′ − δ )) ,γ ( t ) ( t ∈ [ t ′′ − δ , t ′′ ]) . Note that, by the choice of δ , α ′ ( t ) = t ′′ − t ′ t ′′ − t ′ − δ ∈ [1 , ε ] , (2.5) | α − ( t ) − t | = (cid:12)(cid:12)(cid:12)(cid:12) t ′′ − t ′ − δ t ′′ − t ′ ( t − t ′ ) − ( t − t ′ − δ ) (cid:12)(cid:12)(cid:12)(cid:12) (2.6) ≤ | t ′ − t ′ − δ | ∨ | t ′′ − t ′′ − δ | ≤ δ . Now we turn to the estimate. We begin with the following basicestimate: L t ′ ,t ′′ ( m ′ , m ′′ ) − L t ′ ,t ′′ ( m ′ , m ′′ ) ≤ L ( γ ) + L ( γ )(2.7) + L (˜ γ | [ t ′ + δ ,t ′′ − δ ] ) − L ( γ ) + ε. By the choice of ˜ γ , we have L (˜ γ | [ t ′ + δ ,t ′′ − δ ] ) − L ( γ )= 12 Z t ′′ t ′ (cid:26) t ′′ − t ′ t ′′ − t ′ − δ | ˙ γ ( t ) | g ( α − ( t )) − | ˙ γ ( t ) | g ( t ) (cid:27) d t + 12 Z t ′′ t ′ (cid:26) t ′′ − t ′ − δ t ′′ − t ′ R g ( α − ( t )) − R g ( t ) (cid:27) d t. Then the choice of δ together with (2.4), (2.5) and (2.6) yields thatthere is a constant ˆ C > T , K ± , d and C suchthat the right hand side of the last equality is bounded from aboveby ˆ Cε . Moreover, by virtue of the choice of δ and δ , (2.3) yields L t ′ ,t ′ + δ ( m ′ , m ′ ) ∨ L t ′′ − δ ,t ′′ ( m ′′ , m ′′ ) ≤ ε . Thus, by minimizing theright hand side of (2.7) over γ and γ , the left hand side is boundedfrom above by (5 + ˆ C ) ε . We can give the same upper bound to L t ′ ,t ′′ ( m ′ , m ′′ ) − L t ′ ,t ′′ ( m ′ , m ′′ ) in the same manner and hence the as-sertion holds. (cid:3) We fix a bounded open M ⊂ M and δ > K + = K + ( M , δ ) and C = C ( M , δ ) the positive constants appeared inRemark 2.2. Let M := n ( t ′ , m ′ ; t ′′ , m ′′ ) ∈ [0 , T ] × M × [0 , T ] × M : t ′ < t ′′ o . For each ( t ′ , m ′ ; t ′′ , m ′′ ) ∈ M , we denote by Γ t ′ ,t ′′ m ′ ,m ′′ the set of all L -minimizing paths joining ( t ′ , m ′ ) to ( t ′′ , m ′′ ). We further define M δ := (cid:8) ( t ′ , m ′ ; t ′′ , m ′′ ) ∈ M : t ′′ − t ′ > δ (cid:9) , ∆ δ := n ( t ′ , t ′′ ) ∈ [0 , T ] × [0 , T ] : t ′′ − t ′ > δ o , Γ δ := [ ( t ′ ,m ′ ; t ′′ ,m ′′ ) ∈M δ Γ t ′ ,t ′′ m ′ ,m ′′ and ι :Γ δ → ∆ δ × C ([0 , → M )by ι ( γ ) := ( t ′ , t ′′ , b γ ) if γ ∈ Γ t ′ ,t ′′ m ′ ,m ′′ where b γ ∈ C ([0 , → M ) is defined by b γ ( u ) := γ (cid:0) t ′ + u ( t ′′ − t ′ ) (cid:1) for 0 ≤ u ≤ δ by the pull-back distance d Γ δ ( γ, η ) := | t ′ − s ′ | + | t ′′ − s ′′ | + sup ≤ u ≤ ρ g ( T ) ( b γ ( u ) , b η ( u ))for γ ∈ Γ t ′ ,t ′′ m ′ ,m ′′ and η ∈ Γ s ′ ,s ′′ n ′ ,n ′′ . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 17 Lemma 2.13. There are constant c , c > such that, for each ≤ t ′ < t ′′ ≤ T , m ′ , m ′′ ∈ M and γ ∈ Γ t ′ ,t ′′ m ′ ,m ′′ , sup t ′ ≤ t ≤ t ′′ | ˙ γ ( t ) | g ( t ) ≤ c n L t ′ ,t ′′ ( m ′ , m ′′ ) t ′′ − t ′ + dK − o + c . Proof. For each 0 ≤ t ′ < t ′′ ≤ T , m ′ , m ′′ ∈ M and γ ∈ Γ t ′ ,t ′′ m ′ ,m ′′ , theclaimed bound follows from Proposition 2.10(iii) and (iv) with con-stants max { c , c } and max { c , c } as c and c . By virtue of Lemma2.11 (cf. Remark 2.2), we can choose them to be independent of t ′ , t ′′ , m ′ , m ′′ and γ . (cid:3) Proposition 2.14 (Compactness result) . We have the following: (i) Equi-Lipschitz estimate: For each ≤ t ′ < t ′′ ≤ T , m ′ , m ′′ ∈ M , γ ∈ Γ t ′ ,t ′′ m ′ ,m ′′ and t ′ ≤ s ≤ u ≤ t ′′ , ρ g ( T ) ( γ ( s ) , γ ( u )) ≤ const . (cid:16) t ′′ − t ′ + 1 (cid:17) | u − s | , where the constant depends on K − , K + , C but not on s , u , t ′ , t ′′ , m ′ , m ′′ and the choice of γ . (ii) Uniform boundedness: Γ δ is uniformly bounded. (iii) Closedness: ι (Γ δ ) is closed in ∆ δ × C ([0 , → M ) , where thetopology of ∆ δ × C ([0 , → M ) is given by the product of theEuclidean one in ∆ δ and the uniform topology in C ([0 , → M ) .Moreover, Γ δ is compact.Proof. Equi-Lipschitz estimate: For 0 ≤ s < u ≤ T , we have ρ g ( T ) (cid:0) γ ( s ) , γ ( u ) (cid:1) ≤ Z us | ˙ γ ( t ) | g ( T ) d t ≤ e K − T Z us | ˙ γ ( t ) | g ( t ) d t, where we have used Proposition 2.9(i) in the last inequality. Then theequi-Lipschitz estimate follows by Lemma 2.13 and the compactness of M .Uniform boundedness: It is obvious from the Equi-Lipschitz esti-mate.Closedness: Define L : ∆ δ × C ([0 , → M ) → ( −∞ , + ∞ ] by L ( t ′ , t ′′ , c ) := 12 Z t ′′ t ′ n | ˙ e c ( t ) | g ( t ) + R g ( t ) ( e c ( t )) o d t where e c ( t ) := c (cid:16) t − t ′ t ′′ − t ′ (cid:17) for t ′ ≤ t ≤ t ′′ . Then L is lower semicon-tinuous (see e.g., [19, Appendix 1, pp198–201]). By using L , ι (Γ δ ) isexpressed as follows: ι (Γ δ ) = n ( t ′ , t ′′ , c ) : c (0), c (1) ∈ M , t ′′ − t ′ ≥ δ and L ( t ′ , t ′′ , c ) ≤ L t ′ ,t ′′ ( c (0) , c (1)) o This expression yields that ι (Γ δ ) is closed in ∆ δ × C ([0 , → M ).Finally, by combining them with Ascoli’s compactness theorem, weconclude that Γ δ is compact. (cid:3) The L -cut locus. Set M := n ( t ′ , m ′ ; t ′′ , m ′′ ) ∈ [0 , T ] × M × [0 , T ] × M : t ′ < t ′′ o . Definition 2.15 ( L -exponential map and L -cut locus) . (i) L exp t ′ ,t ′′ m ′ : T m ′ M → M is defined by L exp t ′ ,t ′′ m ′ ( v ) := γ ( t ′′ )where the curve γ is the solution to the L -geodesic equation ( ∇ g ( t )˙ γ ( t ) ˙ γ ( t ) − ∇ g ( t ) R g ( t ) − g ( t ) ( ˙ γ ( t ) , · ) = 0 ,γ ( t ′ ) = m ′ , ˙ γ ( t ′ ) = v ∈ T m ′ M. See the Remark 2.3(i) below.(ii) L cut t ′ ,t ′′ m ′ is the set of all points m ′′ ∈ M such that there is morethan one L -minimizing curve γ : [ t ′ , t ′′ ] → M with γ ( t ′ ) = m ′ and γ ( t ′′ ) = m ′′ or there is a v ∈ T m ′ M such that m ′′ = L exp t ′ ,t ′′ m ′ ( v ) and v is a critical point of L exp t ′ ,t ′′ m ′ . We also usethe notation L cut t ′ ,t ′′ := { ( m ′ , m ′′ ) : m ′′ ∈ L cut t ′ ,t ′′ m ′ } .(iii) L cut := n ( t ′ , m ′ ; t ′′ , m ′′ ) ∈ M : m ′′ ∈ L cut t ′ ,t ′′ m ′ o . Remark 2.3. (i) Since our Ricci flow is assumed to be complete, we seethat L exp t ′ ,t ′′ m ′ is well-defined as follows. Given initial data γ ( t ′ ) = m ′ and ˙ γ ( t ′ ) = v , let I ⊂ [ t ′ , T ] be the maximal interval, on which the L -geodesic equation for γ can be solved (Recall that our Ricci flowis defined on [0 , T ]). Since the L -geodesic equation is of the normalform, the standard theory of ODE shows that I is open in [ t ′ , T ]. Onthe other hand, the Gronwall lemma, applied to the first inequality inProposition 2.10(iii) (take u ′ = t ′ and u ′′ = t ), gives an upper boundfor sup t ∈ I | ˙ γ ( t ) | g ( t ) . This implies that { γ ( t ) : t ∈ I } is a bounded set.From the completeness of the metric, γ ( t ) converges as t → sup I ,which turns to show that I is closed in [ t ′ , T ]. Since [ t ′ , T ] is connectedand I = ∅ , we must have I = [ t ′ , T ]. As v is arbitrary, we concludethat L exp t ′ ,t ′′ m ′ is defined on the whole of T m ′ M .(ii) For any L -geodesic γ : [ t ′ , t ′′ ] → M , if ˙ γ ( t ′ ) = 0, then ˙ γ ( t ′′ ) = 0must hold. This can be seen as follows: Let η ( t ) := γ ( t ′′ + t ′ − t ) and e g ( t ) := g ( t ′′ + t ′ − t ) for t ∈ [ t ′ , t ′′ ]. Then η satisfies the differentialequation ∇ e g ( t )˙ η ( t ) ˙ η ( t ) − ∇ e g ( t ) R e g ( t ) + 2Ric e g ( t ) ( ˙ η ( t ) , · ) = 0 COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 19 with η ( t ′ ) = γ ( t ′′ ) and ˙ η ( t ′ ) = − ˙ γ ( t ′′ ). If ˙ γ ( t ′′ ) = 0, then ∇ e g ( t ′ ) R e g ( t ′ ) ( η ( t ′ )) = ∇ g ( t ′′ ) R g ( t ′′ ) ( γ ( t ′′ )) = 0. Therefore η ∗ ( t ) ≡ η ( t ′ ) also satisfies the samedifferential equation as η with the same initial condition. By unique-ness, η ∗ ( t ) ≡ η ( t ) must hold and hence ˙ γ ( t ′ ) = − ˙ η ( t ′′ ) = 0. Thus theassertion holds. Proposition 2.16. (i) For each t ′ < t ′′ and m ′ , m ′′ ∈ M , there is a smooth path γ : [ t ′ , t ′′ ] → M joining m ′ to m ′′ such that γ has the minimal L -length among all such paths (see Lemma 7.27 in [6] ). (ii) For any m ′ ∈ M and t ′ < t ′′ , L cut t ′ ,t ′′ m ′ is closed and of measurezero (see Lemma 7.99 in [6] and Lemma 5 in [20] ). (iii) The L cut is closed. (iv) The function L t ′ ,t ′′ ( m ′ , m ′′ ) is smooth on M\L cut . Remark 2.4. By Proposition 2.1, any L -minimizing curve must be L -geodesic and then Proposition 2.16(i) shows that exp t ′ ,t ′′ m ′ is surjec-tive. Additionally, by (iv), the statements in Proposition 2.2, 2.3 and2.7 hold outside L cut.Since they can be shown by the same arguments as the usual Rie-mannian geometry or L -geometry, we omit the proof of (i) and (iv).The proof of (iii) is along the same line of Lemma 5 in [20]. Proof of Proposition 2.16 (ii). Since the closedness follows from (iii),we prove that L cut t ′ ,t ′′ m ′ is of measure zero. First we decompose L cut t ′ ,t ′′ m ′ = C ∪ C ′ where C is the set of all critical values in L cut t ′ ,t ′′ m ′ of L exp t ′ ,t ′′ m ′ and C ′ := n m ′′ ∈ L cut t ′ ,t ′′ m ′ : m ′′ is a regular value of L exp t ′ ,t ′′ m ′ andthere is more than one L -minimizingcurve joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ). o . By Sard’s theorem, C has measure zero and hence we need only toprove so is also C ′ . For this, consider the map φ : T m ′ M × T m ′ M → R defined by φ ( v, w ) := L ( γ v ) − L ( γ w )where for each v ∈ T m ′ M , the curve γ v : [ t ′ , t ′′ ] → M is given by γ v ( t ) := L exp t ′ ,tm ′ ( v ) , t ′ ≤ t ≤ t ′′ . Then by the first variational formula for L (Proposition 2.1), we have(d φ ) ( v,w ) = (cid:16) (cid:10) ˙ γ v ( t ′′ ) , (d L exp t ′ ,t ′′ m ′ ) v ( · ) (cid:11) g ( t ′′ ) , − (cid:10) ˙ γ w ( t ′′ ) , (d L exp t ′ ,t ′′ m ′ ) w ( · ) (cid:11) g ( t ′′ ) (cid:17) . Therefore by Remark 2.3(ii), the implicit function theorem tells us that N := n ( v, w ) ∈ T m ′ M × T m ′ M : φ ( v, w ) = 0, v = w ;both v and w are regular points of L exp t ′ ,t ′′ m ′ o is a (2 d − T m ′ M × T m ′ M .Take a countable cover { ( U i , ψ i ) } i of M which consists of local coor-dinate neighborhoods and consider the map ξ i : N i := N ∩ (cid:2)(cid:0) L exp t ′ ,t ′′ m ′ (cid:1) − ( U i ) (cid:3) → R d defined by ξ i ( v, w ) := ψ i (cid:0) L exp t ′ ,t ′′ m ′ ( v ) (cid:1) − ψ i (cid:0) L exp t ′ ,t ′′ m ′ ( w ) (cid:1) . Under the identification T m ′′ M ∼ = T ψ i ( m ′′ ) R d for m ′′ ∈ U i , we have(d ξ i ) ( v,w ) = (cid:16) (d L exp t ′ ,t ′′ m ′ ) v , − (d L exp t ′ ,t ′′ m ′ ) w (cid:17) . Again by the implicit function theorem, we see that N ′ i := n ( v, w ) ∈ N i : both (d L exp t ′ ,t ′′ m ′ ) v and (d L exp t ′ ,t ′′ m ′ ) w arenon-singular and L exp t ′ ,t ′′ m ′ ( v ) = L exp t ′ ,t ′′ m ′ ( w ) o is a ( d − T m ′ M × T m ′ M .Now, letting η i : N ′ i ∋ ( v, w ) 7→ L exp t ′ ,t ′′ m ′ ( v ) ∈ M, C ′ is included in the countable union of hypersurfaces η i ( N ′ i ) of M .Therefore it has measure zero. (cid:3) Proof of Proposition 2.16 (iii). Assume that L cut is not closed.Then we can take a sequence ( t ′ i , m ′ i ; t ′′ i , m ′′ i ) ∈ L cut, i = 1 , , . . . whichconverges to some point ( t ′ , m ′ ; t ′′ , m ′′ ) / ∈ L cut. We denote ∆ := { ( t ′ , t ′′ ) ∈ [0 , T ] : t ′ < t ′′ } .Then the map ϕ : ∆ × T M → ([0 , T ] × M ) defined by ϕ (cid:0) t ′ , t ′′ , ( m ′ , v ′ ) (cid:1) := (cid:0) t ′ , m ′ ; t ′′ , L exp t ′ ,t ′′ m ′ ( v ) (cid:1) is non-singular at (cid:0) t ′ , t ′′ , ( m ′ , v ′ ) (cid:1) where v ′ := ˙ γ ( t ′′ ) and γ is theunique L -minimizing curve joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ). Hence bythe inverse function theorem, we can take an open neighborhood U of (cid:0) t ′ , t ′′ , ( m ′ , v ′ ) (cid:1) and an open neighborhood V of ( t ′ , m ′ ; t ′′ , m ′′ ) suchthat ϕ | U : U → V is diffeomorphic. Define v ′ i ∈ T m ′ i M by (cid:0) t ′ i , t ′′ i , ( m ′ i , v ′ i ) (cid:1) = ϕ | − U ( t ′ i , m ′ i ; t ′′ i , m ′′ i ) . We may assume that ( t ′ i , m ′ i ; t ′′ i , m ′′ i ) ∈ V for all i with neglecting finite numbers of points if necessary.Note that, for each (cid:0) t ′ , t ′′ , ( m ′ , v ′ ) (cid:1) ∈ U , v ′ is not a critical pointof L exp t ′ ,t ′′ m ′ : T m ′ M → M . Therefore we can choose w ′ i ∈ T m ′ i M sothat [ t ′ i , t ′′ i ] ∋ t 7→ L exp t ′ ,tm ′ ( w ′ i ) is a L -minimizing geodesic joining( t ′ i , m ′ i ) to ( t ′′ i , m ′′ i ) but v ′ i = w ′ i . Taking a subsequence if necessary,we may assume that (cid:0) t ′ i , t ′′ i , ( m ′ i , w ′ i ) (cid:1) i →∞ → (cid:0) t ′ , t ′′ , ( m ′ , w ′ ) (cid:1) for some COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 21 w ′ ∈ T m ′ M . Since (cid:0) t ′ i , t ′′ i , ( m ′ i , w ′ i ) (cid:1) / ∈ U for all i and ϕ | U : U → V isdiffeomorphic, we must have w ′ = v ′ .Consequently, the curves t 7→ L exp t ′ ,tm ′ ( v ′ ) and t 7→ L exp t ′ ,tm ′ ( w ′ )must be distinct L -minimizing curves joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ),which contradicts to ( t ′ , m ′ ; t ′′ , m ′′ ) / ∈ L cut. Hence L cut is closed. (cid:3) We introduce the notion of L -Jacobi fields. For a smooth curve γ : [ t ′ , t ′′ ] → M , a C -vector field V along γ and t ∈ [ t ′ , t ′′ ], we define alinear form J t ( V ) as follows: J t ( V ) := −∇ g ( t )˙ γ ( t ) ∇ g ( t )˙ γ ( t ) V ( t ) + 12 Hess g ( t ) R g ( t ) ( V ( t ) , · )(2.8) − R g ( t ) ( V ( t ) , ˙ γ ( t )) ˙ γ ( t ) + 2 (cid:0) ∇ g ( t ) V ( t ) Ric g ( t ) (cid:1) ( ˙ γ ( t ) , · )+ 2Ric g ( t ) (cid:0) ∇ g ( t ) V ( t ) ˙ γ ( t ) , · (cid:1) . We call a vector field J along γ an L -Jacobi field if J t ( J ) = 0 for each t ∈ [ t ′ , t ′′ ]. Note that some computation yields the following relationbetween J t and L I γ :(2.9) L I γ ( V, W ) = (cid:10) ∇ g ( t )˙ γ ( t ) V ( t ) , W ( t ) (cid:11) g ( t ) (cid:12)(cid:12)(cid:12) t = t ′′ t = t ′ + Z t ′′ t ′ h J t ( V ) , W ( t ) i g ( t ) d t. Lemma 2.17. Let γ : [ t ′ , t ′′ ] → M be an L -geodesic. Then ˙ γ ( t ′ ) is acritical point of L exp t ′ ,t ′′ γ ( t ′ ) if and only if there is an L -Jacobi field J along γ with J ( t ′ ) = 0 , J ( t ′′ ) = 0 and J .Proof. Let ˙ γ ( t ′ ) be a critical point of L exp t ′ ,t ′′ γ ( t ′ ) . It means that there is anon-zero vector V ∈ T ˙ γ ( t ′ ) ( T γ ( t ′ ) M ) satisfying (d L exp t ′ ,t ′′ γ ( t ′ ) ) ˙ γ ( t ′ ) ( V ) = 0.We consider a variation f ( u, t ) : ( − ε, ε ) × [ t ′ , t ′′ ] → M of γ given by(2.10) f ( u, t ) := L exp t ′ ,tγ ( t ′ ) (cid:0) ˙ γ ( t ′ ) + u ( t − t ′ ) V (cid:1) . Then the vector field J along γ given by J ( t ) := dd u (cid:12)(cid:12) u =0 f ( u, t ) is an L -Jacobi field. Indeed, we can verify it by applying ∇ g ( t )( ∂/∂u ) | u =0 f ( u,t ) tothe L -geodesic equation G t ( f ( u, · )) = 0 for f ( u, · ). Then J ( t ′ ) = 0 bydefinition and J ( t ′′ ) = 0 by the choice of V . In addition, ∇ g ( t ′ )˙ γ ( t ′ ) J ( t ′ ) = V holds (Here we are identifying V with a vector in T γ ( t ′ ) M ). Since V = 0,we have J L -Jacobi field J along γ with J ( t ′ ) = 0, J ( t ′′ ) = 0 and J 0. Then V := ∇ g ( t ′ )˙ γ ( t ′ ) J = 0. Indeed,if V = 0, then J ≡ J t ( J ) = 0. By identifying V with a vector in T ˙ γ ( t ′ ) ( T γ ( t ′ ) M ), we consider a variation f given by (2.10). Then again by the uniqueness of the solution to J t ( J ) = 0,the variational vector field corresponding to f coincides with J . Then J ( t ′′ ) = 0 means (d L exp t ′ ,t ′′ γ ( t ′ ) ) ˙ γ ( t ′ ) ( V ) = 0 and hence the conclusionholds. (cid:3) Proposition 2.18. For each ( t ′ , m ′ ; t ′′ , m ′′ ) ∈ M , the following twoconditions are equivalent: (i) ( t ′ , m ′ ; t ′′ , m ′′ ) / ∈ L cut . (ii) Every L -minimizing curve γ joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ) ex-tends to an L -minimizing curve beyond ( t ′′ , m ′′ ) , that is, thereexist an ε > and a curve e γ : [ t ′ , t ′′ + ε ] → M such that e γ | [ t ′ ,t ′′ ] = γ and e γ is an L -minimizing curve joining its end-points.Proof. Assume that (i) holds. If γ is an L -minimizing curve joining( t ′ , m ′ ) and ( t ′′ , m ′′ ), then γ is unique among such curves and satisfiesthe L -geodesic equation. Since the map T m ′ M ∋ w 7→ L exp t ′ ,t ′′ m ′ ( w ) ∈ M is regular at ˙ γ ( t ′ ), there exists an open neighborhood U of ˙ γ ( t ′ ) suchthat the map is diffeomorphic on U to its image.We extend γ forward in time, to e γ : [ t ′ , t ′′ + ε ] → M by solvingthe same L -geodesic equation ( ε > L cut is closed (Proposition 2.16(iii)), we can assume that e γ [ t ′′ , t ′′ + ε ] ∩ L cut t ′ ,t ′′ + ε e γ ( t ′ ) = ∅ for sufficiently small ε > 0. For each ε ∈ (0 , ε ], we define w ε ∈ T γ ( t ′ ) M to be the tangent vector ˙ γ ε ( t ′ ) of theunique L -minimizing curve γ ε joining ( t ′ , γ ( t ′ )) and ( t ′′ + ε, γ ( t ′′ + ε )). By Proposition 2.16(iv) and Proposition 2.3, L is smooth around( t ′ , m ′ ; t ′′ + ε, γ ( t ′′ + ε )) and | w ε | g ( t ′ ) = 2 ∂L t ′ ,t ′′ + ε ∂t ′ (cid:0) m ′ , γ ( t ′′ + ε ) (cid:1) + R g ( t ′ ) ( m ′ ) , so that { w ε } <ε ≤ ε is a bounded set. Therefore, taking a subsequence,we can assume that w ε converges to a vector w ∈ T ˙ γ ( t ′ ) M as ε ↓ w ε ∈ U for a sufficiently small ε > 0. Since γ ε ( t )smoothly depends on w ε and L is lower semi-continuous (cf. the proofof Proposition 2.14), by using Lemma 2.12, we have L ( γ ) = L t ′ ,t ′′ (cid:0) γ ( t ′ ) , γ ( t ′′ ) (cid:1) = lim ε → L t ′ ,t ′′ (cid:0) γ ε ( t ′ ) , γ ε ( t ′′ + ε ) (cid:1) = lim ε → L ( γ ε ) ≥ L ( γ w ) ≥ L ( γ ) , where γ w : [ t ′ , t ′′ ] → M is the L -geodesic of initial conditions γ w ( t ′ ) = m ′ and ˙ γ w ( t ′ ) = w . It clearly holds that γ w ( t ′′ ) = m ′′ . Therefore, γ w is also an L -minimizing curve joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ). By theuniqueness, we have γ = γ w , so that ˙ γ ( t ′ ) = w . Thus w ε ∈ U for asufficiently small ε > COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 23 Now, for a sufficiently small ε > 0, we have ˙ γ ( t ′ ), w ε ∈ U and L exp t ′ ,t ′′ + εγ ( t ′ ) ( ˙ γ ( t ′ )) = e γ ( t ′′ + ε ) = γ w ( t ′′ + ε ) = L exp t ′ ,t ′′ + εγ ( t ′ ) ( w ε ) , which implies ˙ γ ( t ′ ) = w ε , so that e γ = γ ε . Hence the curve e γ is uniquely L -minimizing.Conversely, assume that (ii) holds. Suppose that γ and η are two L -minimizing curves joining ( t ′ , m ′ ) and ( t ′′ , m ′′ ). By the assumption(ii), γ has an L -minimizing extension e γ : [ t ′ , t ′′ + ε ] → M . Then thepiecewise smooth curve c defined by c ( t ) := (cid:26) η ( t ) if t ∈ [ t ′ , t ′′ ], e γ ( t ) if t ∈ [ t ′′ , t ′′ + ε ],which is the concatenation of η and e γ | [ t ′ ,t ′′ + ε ] , must be also L -minimizing.A standard variational argument shows that the curve c ( t ) becomes C at t = t ′′ and must be L -geodesic (so that c is found to be smooth).By the uniqueness result (with respect to the initial conditions at time t ′′ + ε ) of the ODE theory, we must have e γ = c , in particular, wehave γ = η . Therefore, the L -minimizing geodesic joining ( t ′ , m ′ ) and( t ′′ , m ′′ ) must be unique.Next, we show that L exp t ′ ,t ′′ m ′ is not singular at ˙ γ ( t ′ ) ∈ T m ′ M . Sup-pose on the contrary, that ˙ γ ( t ′ ) is a critical point of L exp t ′ ,t ′′ m ′ . Thenthere is an L -Jacobi field along γ with J ( t ′ ) = 0, J ( t ′′ ) = 0 and J ∇ g ( t ′′ ) γ ( t ′′ ) J ( t ′′ ) = 0 holds (cf. the proof ofLemma 2.17). We take an L -minimizing extension e γ : [ t ′ , t ′′ + ε ] → M of γ and extend J to a piecewise smooth vector field on [ t ′ , t ′′ + ε ] (whichwe denote again by J ) by requiring that J | [ t ′′ ,t ′′ + ε ] ≡ 0. We furthertake a vector field V along e γ with V ( t ′ ) = 0, V ( t ′′ ) = −∇ g ( t ′′ )˙ γ ( t ′′ ) J ( t ′′ − )and V ( t ′′ + ε ) = 0, and then consider any proper variation g ( u, t ) :( − δ, δ ) × [ t ′ , t ′′ ] → M of e γ , with the variational vector field W = J + aV ( a > W vanishes at the endpoints of e γ , such a proper variationexists. By a piecewise use of the second variational formula (Proposi-tion 2.4) together with (2.9) and the symmetry of L I γ , we have( δ W δ W L )( e γ ) = − a |∇ g ( t ′′ )˙ γ ( t ′′ ) J ( t ′′ − ) | g ( t ′′ ) + a ( δ V δ V L )( e γ )which is negative for sufficiently small a > 0. On the other hand, since e γ is L -minimizing and g is proper, it must hold that L ( e γ ) = L ( g (0 , · )) ≤ L ( g ( u, · )) , for each u ∈ ( − δ, δ )which implies ( δ W δ W L )( e γ ) = (d / d u ) | u =0 L ( g ( u, · )) ≥ 0. This is acontradiction. Therefore ˙ γ ( t ′ ) is not a critical point of L exp t ′ ,t ′′ m ′ .Hence we have proved (ii) ⇒ (i). (cid:3) Finally we study relations in the time-reversal and L -cut locus. Let e g ( τ ) = g ( T − τ ). Then e g ( τ ) evolves under the backward Ricci flow ∂∂τ e g ( τ ) = 2Ric e g ( τ ) . For τ ′ < τ ′′ , we consider the corresponding functional e L ( c ) := 12 Z τ ′′ τ ′ (cid:8) | ˙ c ( τ ) | e g ( τ ) + R e g ( τ ) ( c ( τ )) (cid:9) d τ, where c : [ τ ′ , τ ′′ ] → M is any curve. For x ′ ∈ M we define the e L -exponential map e L exp τ ′ ,τ ′′ x ′ : T x ′ M → M by e L exp τ ′ ,τ ′′ x ′ ( w ) := η ( τ ′′ )where η is the solution to the e L -geodesic equation ( ∇ e g ( τ )˙ η ( τ ) ˙ η ( τ ) − ∇ e g ( τ ) R e g ( τ ) + 2Ric e g ( τ ) ( ˙ η ( τ ) , · ) = 0 ,η ( τ ′ ) = x ′ , ˙ η ( τ ′ ) = w. (2.11)One can see that this is actually the Euler-Lagrange equation for e L .Take τ ′ , τ ′′ , t ′ , t ′′ ∈ [0 , T ] with τ ′ < τ ′′ , τ ′ = T − t ′′ and t ′′ − t ′ = τ ′′ − τ ′ .For a curve η : [ τ ′ , τ ′′ ] → M , we define γ : [ t ′ , t ′′ ] → M by γ ( t ′ + s ) = η ( τ ′′ − s ) for 0 ≤ s ≤ t ′′ − t ′ = τ ′′ − τ ′ . We call γ the time-reversal of η .By definition, e L ( η ) = L ( γ ) holds. In addition, by comparing (2.11)with (2.1), we can easily show that γ is an L -geodesic if and only if η is an e L -geodesic. Proposition 2.19. Let τ ′ , τ ′′ , t ′ , t ′′ ∈ [0 , T ] with τ ′ < τ ′′ , τ ′ = T − t ′′ and t ′′ − t ′ = τ ′′ − τ ′ . (i) There is more than one L -minimizing curve joining ( t ′ , x ) and ( t ′′ , y ) iff there is more than one e L -minimizing curve joining ( τ ′ , y ) and ( τ ′′ , x ) . (ii) Let η : [ τ ′ , τ ′′ ] → M be an e L -minimizing curve and γ its time-reversal. Then the vector ˙ γ ( t ′ ) is a critical point of L exp t ′ ,t ′′ γ ( t ′ ) iff ˙ η ( τ ′ ) is a critical point of e L exp τ ′ ,τ ′′ η ( τ ′ ) . (iii) Define e L cut τ ′ ,τ ′′ x ′ as the set of all points x ′′ ∈ M such thatthere is more than one e L -minimizing curve joining ( τ ′ , x ′ ) and ( τ ′′ , x ′′ ) or there is a w ∈ T x ′ M such that x ′′ = e L exp τ ′ ,τ ′′ x ′ ( w ) and w is a critical point of e L exp τ ′ ,τ ′′ x ′ . We also define e L cut τ ′ ,τ ′′ := (cid:8) ( x ′ , x ′′ ) : x ′′ ∈ e L cut τ ′ ,τ ′′ x ′ (cid:9) . Then the map M × M ∋ ( x, y ) ( y, x ) ∈ M × M gives anisomorphism between L cut t ′ ,t ′′ and e L cut τ ′ ,τ ′′ . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 25 (iv) ( x ′ , x ′′ ) / ∈ e L cut τ ′ ,τ ′′ if and only if e L -minimizing curve joining ( τ ′ , x ′ ) and ( τ ′′ , x ′′ ) can be extended beyond ( τ ′′ , x ′′ ) with keepingits minimality.Proof. The claim (i) is obvious by remarks just before Proposition 2.19.For (ii), we introduce the notion of e L -Jacobi field: For η : [ τ ′ , τ ′′ ] → M , e J is an e L -Jacobi field if f J τ ( V ) = 0, where a linear form f J τ is de-fined by replacing γ ( t ), ˙ γ ( t ) and g ( t ) in (2.8) with η ( τ ), ˙ η ( τ ) and e g ( τ )respectively and changing the sign of all terms involving the Ricci cur-vature. Then the criticality of e L -exponential map is also characterizedby e L -Jacobi fields as in Lemma 2.17 by the same argument. Moreover,a vector field e J along η is an e L -Jacobi field if and only if a vector field J along the time-reversal of η given by J ( t ′ + s ) := e J ( τ ′′ − s ) is an L -Jacobi field. Then the conclusion easily follows by combining theseobservations with Lemma 2.17. The assertion (iii) follows immediatelyfrom (i) and (ii). The assertion (iv) can be proved in the same way asProposition 2.18. (cid:3) Construction of a coupled Brownian motions in theabsence of L -cut locus In this section, we will show Theorem 1.1 under the assumption that L cut is empty where L cut is defined in section 2.3.Under L cut = ∅ , L t ′ ,t ′′ ( m ′ , m ′′ ) is a smooth function of ( t ′ , m ′ ; t ′′ , m ′′ )(Theorem 2.16(iv)), and for each m ′ , m ′′ ∈ M and t ′ < t ′′ , we can take aunique minimizer for L joining ( t ′ , m ′ ) to ( t ′′ , m ′′ ). Therefore, for eachpair of points, the space-time parallel transport as a map between theirtangent spaces is uniquely determined. In the following, we construct acoupled Brownian motions by space-time parallel transport, introducedin Definition 2.6.Let e , e , . . . , e d ∈ R d be the canonical basis of R d . We denote by π : F ( M ) → M the frame bundle over M and by π : O g ( t ) ( M ) → M theorthonormal frame bundle with respect to the metric g ( t ). Note that O g ( t ) ( M ) varies along our Ricci flow but not for F ( M ). We canonicallyhave the map defined by F ( M ) m × R d ∋ ( u, x ) u.x := u ( x ) ∈ T m M, where F ( M ) m stands for the fiber at m of F ( M ).Recall our notations on time parameters τ ′ ( s ), τ ′′ ( s ) given in section1: We sometimes drop s and simply write τ ′ or τ ′′ . Let ( U s , V s ) ≤ s ≤ S = ( U s ( u ′ ) , V s ( v ′′ )) ≤ s ≤ S be the solution to the SDE d U s = √ H g ( τ ′ ) i ( U s ) ◦ d W is − d X α,β =1 ∂g∂t ( τ ′ ) (cid:16) U s .e α , U s .e β (cid:17) V α,β ( U s )d s, d V s = √ H g ( τ ′′ ) i ( V s ) ◦ d B is − d X α,β =1 ∂g∂t ( τ ′′ ) (cid:16) V s .e α , V s .e β (cid:17) V α,β ( V s )d s, d B s = V − s // τ ′ ,τ ′′ πU s ,πV s U s d W s , starting from ( U , V ) = ( u ′ , v ′′ ) ∈ O g ( t ′ ) ( M ) m ′ × O g ( t ′′ ) ( M ) m ′′ , where( H g ( t ) i ) di =1 is the system of canonical horizontal vector fields on F ( M )associated with g ( t ), and ( V α,β ) dα,β =1 is the system of canonical verticalvector fields , each of which is defined by V α,β f ( u ) = dd ε (cid:12)(cid:12)(cid:12)(cid:12) ε =0 f ( u ◦ e εe α ⊗ e β ) , f ∈ C ∞ ( F ( M )) . It is known that the random variable ( U s , V s ) takes its value in O g ( τ ′ ( s )) ( M ) × O g ( τ ′′ ( s )) ( M ) for every 0 ≤ s ≤ S (see [2, Proposition 1.1]). We put( X s , Y s ) := ( πU s , πV s ).The next statement is an Itˆo formula for the process ( X, Y ) =( X s , Y s ) ≤ s ≤ S . We omit the proof because it is straightforward. Proposition 3.1. For any smooth function f ( s, m ′ , m ′′ ) on [0 , ∞ ) × M × M , we have d f ( s, X s , Y s )= n ∂f∂s ( s, X s , Y s ) + d X i =1 h Hess g ( τ ′ ) ⊕ g ( τ ′′ ) f ( s, · , · ) i(cid:16) U s .e i ⊕ V s .e ∗ i , U s .e i ⊕ V s .e ∗ i (cid:17)o d s + [ U s .e i ⊕ V s .e ∗ i ] f ( s, · , · ) • d W is , where e ∗ i := V − s // τ ′ ,τ ′′ X s ,Y s U s .e i ∈ R d for each i = 1 , , . . . , d and • d W is means the Itˆo integral. Corollary 3.2. Suppose L cut = ∅ . Then we have the following: d L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s )= − n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o(cid:12)(cid:12)(cid:12) ( t ′ ,t ′′ )=( τ ′ ( s ) ,τ ′′ ( s )) ( X s , Y s )d s + d X i =1 h Hess g ( τ ′ ) ⊕ g ( τ ′′ ) L τ ′ ,τ ′′ i(cid:16) U s .e i ⊕ V s .e ∗ i , U s .e i ⊕ V s .e ∗ i (cid:17) d s + [ U s .e i ⊕ V s .e ∗ i ] L τ ′ ,τ ′′ • d W is . Since L cut = ∅ , by using the result in Proposition 2.7, the sto-chastic process L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is a semi-martingale whose boundedvariation part is non-positive. Therefore we can conclude that s COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 27 L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is a supermartingale after we prove the integrabil-ity. For proving the integrability, we consider a family of functions ϕ n ∈ C ( R ) with ϕ n ( x ) ↑ x as n → ∞ for each x ∈ R . Sup-pose in addition that ϕ n is nondecreasing, concave and bounded fromabove. Since L is bounded from below, we can easily show that ϕ n ( L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s )) is a supermartingale. Then the integrabilitywill be ensured by the monotone convergence theorem and the fact L τ ′ (0) ,τ ′′ (0)0 ( X , Y ) is deterministic. The proof of the integrability inthe next section will go along the same idea but we will show it to-gether with the rigorous proof of Theorem 1.1.4. Supermartingale property of a coupled Brownianmotion in the presence of L -cut locus Coupling via approximation by geodesic random walks. For the construction of a suitable coupling of Brownian motions in thepresence of L -cut locus, we use the approximation by geodesic randomwalks to avoid the technical difficulty coming from the singularity of L . Indeed L is smooth outside L cut (Proposition 2.16(iv)) but noton L cut. To carry out this procedure, we will rely on some basicproperties of L cut summarized in section 2.3.We fix a measurable section (or selection) γ : ( t ′ , m ′ ; t ′′ , m ′′ ) γ t ′ ,t ′′ m ′ ,m ′′ of minimal L -geodesics, where, the measurability is with respect tothe Borel σ -field generated by the uniform topology on the path space.Since L -minimizing curves with fixed endpoints form a compact set(Proposition 2.14), the existence of such a section is ensured by ameasurable selection theorem (e.g. [3, Theorem 6.9.6] and Proposition2.14). We further fix a measurable section σ ( t, · ) : M → O g ( t ) ( M ) of g ( t )-orthonormal frame bundle for each t ∈ [ t ′ , t ′ ].To construct a geodesic random walk, we prepare an R d -valued i.i.d.sequence ( λ n ) ∞ n =1 , each of which is uniformly distributed on the unitball in R d centered at the origin. Since it will be needed when workingwith these λ ’s, we shall summarize necessary formulae as follows. Weomit the proof because it is obvious or an easy consequence of thedivergence theorem. Lemma 4.1. Let V be an n -dimensional real Euclidean space. Let ℓ : V → R be a linear function and α : V × V → R be a symmetric -form on V . Let B n is the unit ball in V centered at origin. Then (i) Z B n ℓ ( x )d x = 0 and Z B n α ( x, x )d x = vol( B n ) n + 2 tr α, where in the last equality, we have naturally regarded α as thelinear homomorphism V → V ∗ ∼ = V . Suppose further that we are given another n -dimensional real vectorspace W , a linear function f : V ⊕ W → R , a symmetric -form β on V ⊕ W and a linear homomorphism A : V → W . Then (ii) Z B n f ( x ⊕ Ax )d x = 0 and (iii) Z B n β (cid:0) x ⊕ Ax, x ⊕ Ax (cid:1) d x = vol( B n ) n + 2 n X i =1 β (cid:0) e i ⊕ Ae i , e i ⊕ Ae i (cid:1) where ( e i ) ni =1 is an any orthonormal basis of V . Now for each m ′ , m ′′ ∈ M and ε > 0, let us construct a coupling( X εs , Y εs ) ≤ s ≤ S of geodesic random walks starting from ( m ′ , m ′′ ) by X εs := exp g ( τ ′ (0)) X ε (cid:16) √ sε σ ( τ ′ (0) , X ε ) . √ d + 2 λ (cid:17) ,Y εs := exp g ( τ ′′ (0)) Y ε (cid:16) √ sε // τ ′ (0) ,τ ′′ (0) X ε ,Y ε σ ( τ ′ (0) , X ε ) . √ d + 2 λ (cid:17) , for 0 ≤ s ≤ s ,... X εs := exp g ( τ ′ ( s n )) X εsn (cid:16) √ s − √ s n ε σ ( τ ′ ( s n ) , X εs n ) . √ d + 2 λ n +1 (cid:17) ,Y εs := exp g ( τ ′′ ( s n )) Y εsn (cid:16) √ s − √ s n ε // τ ′ ( s n ) ,τ ′′ ( s n ) X εsn ,Y εsn σ ( τ ′ ( s n ) , X εs n ) . √ d + 2 λ n +1 (cid:17) for s n ≤ s ≤ s n +1 ...where X ε := m ′ , Y ε := m ′′ and s n := ( nε ) ∧ S for each n = 0 , , , . . . .From Lemma 4.1, we see that the factor √ d + 2 is the normalizationconstant in the sense of that ( d + 2) E (cid:2) λ n ⊗ λ n (cid:3) = id R d . We shall give a remark here. From the definition, the random curves s X εs and s Y εs are clearly piecewise smooth and then we see that X εs n and Y εs n are σ (cid:0) λ k : 1 ≤ k ≤ n (cid:1) -measurable and ˙ X εs n + and ˙ Y εs n + are σ (cid:0) λ k : 1 ≤ k ≤ n + 1 (cid:1) -measurable but not σ (cid:0) λ k : 1 ≤ k ≤ n (cid:1) -measurable.As shown in [16], each of X ε = ( X εs ) ≤ s ≤ S and Y ε = ( Y εs ) ≤ s ≤ S converges in law to g ( τ ′ ( s ))-Brownian motion starting from m ′ and g ( τ ′′ ( s ))-Brownian motion starting from m ′′ respectively. As a result,the collection { ( X ε , Y ε ) } ε> of couplings forms a tight family and hencewe can find a convergent subsequence of ( X ε , Y ε ) ε> . We fix such a sub-sequence and denote the subsequence by the same notation ( X ε , Y ε ) ε> for simplicity. We denote the limit by ( X, Y ) = ( X s , Y s ) ≤ s ≤ S .4.2. Supermartingale property. Let G be the trivial σ -field and G n := σ (cid:0) λ k : 1 ≤ k ≤ n (cid:1) for each n = 1 , , · · · . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 29 Proposition 4.2. Set Λ εs := L τ ′ ( s ) ,τ ′′ ( s )0 ( X εs , Y εs ) . For each relativelycompact open set M in M including X ε = m ′ and Y ε = m ′′ , there is ε > which depends only on M such that the following holds: Foreach < ε ≤ ε , there are a family ( Q εn ) ∞ n =1 of random variables and adeterministic constant δ ( ε ) such that (4.1) Λ εs n +1 ≤ Λ εs n + εζ εn +1 + ε Σ εn +1 + Q εn +1 with the estimate (4.2) X n : s n <σ M ( X ε ,Y ε ) ∧ S Q εn ≤ δ ( ε ) → as ε → .where ζ εn and Σ εn are G n -measurable and integrable random variablessuch that (4.3) E [ ζ εn | G n − ] = 0 and Σ εn := E [ Σ εn | G n − ] ≤ for each n ≥ , and σ M ( w ′ , w ′′ ) := inf (cid:8) s > w ′ ( s ) , w ′′ ( s )) / ∈ M × M (cid:9) for each ( w ′ , w ′′ ) ∈ C ([0 , S ] → M × M ) . Remark 4.1. Intuitively, it is clear that the difference inequality (4.1)comes from the Taylor expansion with respect to ε . Therefore, when( τ ′ ( s n ) , X εs n ; τ ′′ ( s n ) , Y εs n ) / ∈ L cut, ζ εn +1 = (cid:2) ε ˙ X εs n + ⊕ ε ˙ Y εs n + (cid:3) L τ ′ ( s n ) ,τ ′′ ( s n )0 Σ εn +1 = − n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o(cid:12)(cid:12)(cid:12) ( t ′ ,t ′′ )=( τ ′ ( s n ) ,τ ′′ ( s n )) ( X εs n , Y εs n )+ 12 (cid:2) Hess g ( τ ′ ( s n )) ⊕ g ( τ ′′ ( s n )) L τ ′ ( s n ) ,τ ′′ ( s n )0 (cid:3)(cid:0) ε ˙ X εs n + ⊕ ε ˙ Y εs n + , ε ˙ X εs n + ⊕ ε ˙ Y εs n + (cid:1) . However, we must avoid using this expression since L cut = ∅ . Proof. If L cut = ∅ , the Taylor expansion with respect to the param-eter ε easily yields the desired estimate. Thus we will modify theargument by taking the presence of L cut into account. We begin withdetermining Q εn +1 which enjoys (4.1) and we verify (4.2) after that. Letus define K ⊂ ([0 , S ] × M ) by K := ( u i , m i ) i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( u , m ) , ( u , m ) ∈ [0 , S ] × M ,u − u ≥ t ′′ − t ′ ,u = ( u + u ) / ,L u ,u ( m , m )= L u ,u ( m , m ) + L u ,u ( m , m ) . By Proposition 2.9 (iii), there are constants c , c > u i , m i ) i =1 ∈ K , ρ g ( t ′ ) ( m , m ) + ρ g ( t ′′ ) ( m , m ) ≤ c ( L u ,u ( m , m ) + L u ,u ( m , m )) + c = c L u ,u ( m , m ) + c Since the last quantity does not depend on m and continuously de-pends on ( m , t ; m , t ) which moves on a compact set, there is abounded set D , which is possibly larger than M , such that ( u , m ) ∈ D holds for any ( u i , m i ) i =1 ∈ K . Therefore K is compact. Take( u i , m i ) i =1 ∈ K . Then ( u , m ) must be on a minimal L -geodesicjoining ( u , m ) and ( u , m ). We denote it by γ . Since u = u by thedefinition of K , both γ | [ u ,u ] and γ | [ u ,u ] are not constant as a space-time curve (It means that ( t, γ ( t )) is not constant both on [ u , u ] andon [ u , u ]). Hence we can extend γ | [ u ,u ] and γ | [ u ,u ] with keepingtheir minimalities. It implies ( u i , m i ; u i +1 , m i +1 ) / ∈ L cut for i = 1 , j = 1 , 2, let p j be a projection from K given by p j (( u i , m i ) i =1 ) = ( u j , m j ; u j +1 , m j +1 ).Then ( p j ( K )) j =1 , are compact and away from L cut. Let us define acompact set K ε ⊂ ([0 , S ] × M ) by A : = ( u ′ , m ′ , v ′ ; u ′′ , m ′′ , v ′′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( u ′ , m ′ ; u ′′ , m ′′ ) ∈ p ( K ) ∪ p ( K ) ,v ′ ∈ T m ′ M, v ′′ ∈ T m ′′ M, k v ′ k g ( u ′ ) = k v ′′ k g ( u ′′ ) = p d + 2) ,K ε : = n (cid:16) u ′ + a, exp g ( τ ′ ( u ′ )) m ′ ( εav ′ ); u ′′ + a, exp g ( τ ′′ ( u ′′ )) m ′′ ( εav ′′ ) (cid:17)(cid:12)(cid:12)(cid:12) ( u ′ , m ′ , v ′ ; u ′′ , m ′′ , v ′′ ) ∈ A, a ∈ [0 , o . Since L cut is closed by Proposition 2.16(iii), there is ε > K ε ∩ L cut = ∅ when ε ≤ ε . Note that the map( t ′ , m ′ ; t ′′ , m ′′ ) (cid:18) t ′ + t ′′ , γ t ′ ,t ′′ m ′ ,m ′′ (cid:18) t ′ + t ′′ (cid:19)(cid:19) is measurable.Let ε < ε . For simplicity of notations, we denote the “midpoint”of ( τ ′ ( s n ) , X εs n ) and ( τ ′′ ( s n ) , Y εs n ), and the associated variational vectorby the following: ˆ τ : = τ ′ ( s n ) + τ ′′ ( s n )2 , ˆ X : = γ τ ′ ( s n ) ,τ ′′ ( s n ) X εsn ,Y εsn (ˆ τ ) , ˆ V : = // τ ′ ( s n ) , ˆ τX εsn , ˆ X ˙ X εs n + . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 31 Then we can easily verify the following: (cid:16) τ ′ ( s n ) , X εs n ; ˆ τ , ˆ X ; τ ′′ ( s n ) , Y εs n (cid:17) ∈ K, (cid:16) τ ′ ( s n ) , X εs n , ε ˙ X εs n + ; ˆ τ , ˆ X, ε ˆ V (cid:17) ∈ K ε , (cid:16) ˆ τ , ˆ X, ε ˆ V ; τ ′′ ( s n ) , Y εs n , ε ˙ Y εs n + (cid:17) ∈ K ε . Since K ε ∩ L cut = ∅ , the Taylor expansion yields L τ ′ ( s n +1 ) , ˆ τ + ε ( X εs n +1 , exp g (ˆ τ )ˆ X ( ε ˆ V ))(4.4) ≤ L τ ′ ( s n ) , ˆ τ ( X εs n , ˆ X ) + ε (cid:2) ε ˙ X εs n + ⊕ ε ˆ V (cid:3) L τ ′ ( s n ) , ˆ τ − ε (cid:2) ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ (cid:3)(cid:12)(cid:12) ( t ′ ,t ′′ )=( τ ′ ( s n ) , ˆ τ ) ( X εs n , ˆ X )+ ε h Hess g ( τ ′ ( s n )) ⊕ g (ˆ τ ) L τ ′ ( s n ) , ˆ τ i(cid:0) ε ˙ X εs n + ⊕ ε ˆ V , ε ˙ X εs n + ⊕ ε ˆ V (cid:1) + o ( ε )and L ˆ τ + ε ,τ ′′ ( s n +1 )0 (exp g (ˆ τ )ˆ X ( ε ˆ V ) , Y εs n +1 )(4.5) ≤ L ˆ τ,τ ′′ ( s n )0 ( ˆ X, Y εs n ) + ε (cid:2) ε ˆ V ⊕ ε ˙ Y εs n + (cid:3) L ˆ τ ,τ ′′ ( s n )0 − ε (cid:2) ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ (cid:3)(cid:12)(cid:12) ( t ′ ,t ′′ )=(ˆ τ,τ ′′ ( s n )) ( ˆ X, Y εs n )+ ε h Hess g (ˆ τ ) ⊕ g ( τ ′′ ( s n )) L ˆ τ,τ ′′ ( s n )0 i(cid:0) ε ˆ V ⊕ ε ˙ Y εs n + , ε ˆ V ⊕ ε ˙ Y εs n + (cid:1) + o ( ε ) . Note that the two remainder terms o ( ε ) appeared in the last equalitiesconsist of higher derivatives of L on K ε . We denote the sum ofthese two remainder terms by Q εn +1 . Since K ε is compact, Q εn +1 iscontrolled uniformly in the position of ( X εs n , Y εs n ) and n as long as s n <σ M ( X ε , Y ε ). It means that there is a constant ˜ δ ( ε ) being independentof ( X εs n , Y εs n ) and n such that Q εn +1 ≤ ˜ δ ( ε ) and ˜ δ ( ε ) /ε → 0. Thetriangular inequality for L and the definition of (ˆ τ , ˆ X ) together with(4.4) and (4.5) yield L τ ′ ( s n +1 ) ,τ ′′ ( s n +1 )0 ( X εs n +1 , Y εs n +1 ) − L τ ′ ( s n ) ,τ ′′ ( s n )0 ( X εs n , Y εs n ) ≤ (cid:16) L τ ′ ( s n +1 ) , ˆ τ + ε ( X εs n +1 , exp g (ˆ τ )ˆ X ( ε ˆ V )) − L τ ′ ( s n ) , ˆ τ ( X εs n , ˆ X ) (cid:17) + (cid:16) L ˆ τ + ε ,τ ′′ ( s n +1 )0 (exp g (ˆ τ )ˆ X ( ε ˆ V ) , Y εs n +1 ) − L ˆ τ,τ ′′ ( s n )0 ( ˆ X, Y εs n ) (cid:17) = εζ εn +1 + ε Σ εn +1 + Q εn +1 , where random variables ζ εn +1 and Σ εn +1 are defined by ζ εn +1 := [ ε ˙ X εs n + ⊕ ε ˆ V ] L τ ′ ( s n ) , ˆ τ + [ ε ˆ V ⊕ ε ˙ Y εs n + ] L ˆ τ,τ ′′ ( s n )0 andΣ εn +1 := − n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o(cid:12)(cid:12)(cid:12) ( t ′ ,t ′′ )=( τ ′ ( s n ) , ˆ τ ) ( X εs n , ˆ X )+ 12 (cid:2) Hess g ( τ ′ ( s n )) ⊕ g (ˆ τ ) L τ ′ ( s n ) , ˆ τ (cid:3)(cid:0) ε ˙ X εs n + ⊕ ε ˆ V , ε ˙ X εs n + ⊕ ε ˆ V (cid:1) − n ∂L t ′ ,t ′′ ∂t ′ + ∂L t ′ ,t ′′ ∂t ′′ o(cid:12)(cid:12)(cid:12) ( t ′ ,t ′′ )=(ˆ τ ,τ ′′ ( s n )) ( ˆ X, Y εs n )+ 12 (cid:2) Hess g (ˆ τ ) ⊕ g ( τ ′′ ( s n )) L ˆ τ,τ ′ ( s n )0 (cid:3)(cid:0) ε ˆ V ⊕ ε ˙ Y εs n + , ε ˆ V ⊕ ε ˙ Y εs n + (cid:1) . Thus (4.1) holds. Since { k ∈ N : s k < σ M ( X ε , Y ε ) ∧ S } ≤ Sε − , we have X n : s n <σ M ( X ε ,Y ε ) ∧ S Q εn +1 ≤ Sε − ˜ δ ( ε ) → ε → 0. It asserts (4.2).Finally, we prove the required properties for ζ εn and Σ εn . The mea-surability are obvious. The integrabilities hold because X εs n , ˆ X and Y εs n lie on a bounded domain in M . Finally (4.3) follows from Lemma 4.1and Proposition 2.7 since λ n is independent of G n − . (cid:3) Let ( F s ) ≤ s ≤ S be the filtration defined by F s := σ (cid:0) ( X u , Y u ) : 0 ≤ u ≤ s (cid:1) , ≤ s ≤ S and set ( F εs n ) ∞ n =0 by F ε := the trivial σ -field, and F εs n := σ (cid:0) ( X εs k , Y εs k ) : k = 1 , , . . . , n (cid:1) for each n = 1 , , . . . . Theorem 4.3. Set Λ s := L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) . Then Λ s is integrable foreach s ∈ [0 , S ] and for each u ≤ s we have E [Λ s |F u ] ≤ Λ u , that is, Λ = (Λ s ) ≤ s ≤ S is an ( F s ) ≤ s ≤ S -supermartingale.Proof. Take η > M ⊂ M sufficiently large so that P [ σ M ( X, Y ) ≤ s ] ≤ η . We first claim that foreach β > C = C ( β ) > M and η such that(4.6) E (cid:2) ( β ∧ Λ s ∧ σ M ( X,Y ) − β ∧ Λ u ∧ σ M ( X,Y ) ) F (cid:3) ≤ Cη for each nonnegative bounded F u -measurable random variable F . Forproving (4.6), we may assume that F is of the form F = f ( Z u , . . . , Z u n ) COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 33 where f is a nonnegative and bounded continuous function on ( M ) n , Z u = ( X u , Y u ) and 0 ≤ u < · · · < u n ≤ u .By Proposition 2.9 (iii), we have E (cid:2) ( β ∧ Λ s ∧ σ M ( X,Y ) − β ∧ Λ u ∧ σ M ( X,Y ) ) F (cid:3) ≤ E (cid:2) ( β ∧ Λ s − β ∧ Λ u ) F : σ M ( X, Y ) > s (cid:3) + C η for some constant C > M but may dependon β (Indeed, one can take C = β + ( dK − T ) / u ] ε := sup (cid:8) ε n : n ∈ N , ε n ≤ u (cid:9) and Z ε := ( X ε , Y ε ). Since { w | σ M ( w ) > s } is open, E (cid:2) ( β ∧ Λ s − β ∧ Λ u ) F : σ M ( X, Y ) > s (cid:3) ≤ lim inf ε → E (cid:2) ( β ∧ Λ εs − β ∧ Λ εu ) f ( Z εu , . . . , Z εu n ) : σ M ( X ε , Y ε ) > s (cid:3) = lim inf ε → E (cid:2) ( β ∧ Λ ε [ s ] ε − β ∧ Λ ε [ u ] ε ) f ( Z ε [ u ] ε , . . . , Z ε [ u n ] ε ) : σ M ( X ε , Y ε ) > s (cid:3) , where the last equality follows from the continuity of L τ ′ ,τ ′′ and f .Then E (cid:2) ( β ∧ Λ ε [ s ] ε − β ∧ Λ ε [ u ] ε ) f ( Z ε [ u ] ε , . . . , Z ε [ u n ] ε ) : σ M ( X ε , Y ε ) > s (cid:3) ≤ E (cid:2) ( β ∧ Λ ε [ s ] ε ∧ [ σ M ( X ε ,Y ε )] ε − β ∧ Λ ε [ u ] ε ∧ [ σ M ( X ε ,Y ε )] ε ) f ( Z ε [ u ] ε , . . . , Z ε [ u n ] ε ) (cid:3) + C P [ σ M ( X ε , Y ε ) ≤ s ] . For the second term of the right hand side,lim sup ε → P [ σ M ( X ε , Y ε ) ≤ s ] ≤ η holds since { w | σ M ( w ) ≤ s } is closed. Let us estimate the first term.Since Proposition 4.2 ensures Σ εi ≤ i = 1 , , . . . , Proposition4.2 together with the conditional Jensen inequality yields E (cid:2) ( β ∧ Λ ε [ s ] ε ∧ [ σ M ( X ε ,Y ε )] ε ) f ( Z ε [ u ] ε , . . . , Z ε [ u n ] ε ) (cid:3) ≤ E " β ∧ ( E h [ s ] ε ∧ [ σ M ( X ε ,Y ε )] ε X i =[ u ] ε ∧ [ σ M ( X ε ,Y ε )] ε +1 εζ εi + ε (Σ εi − Σ εi ) + Q εi (cid:12)(cid:12) F ε [ u ] ε i + Λ ε [ u ] ε ∧ [ σ M ( X ε ,Y ε )] ε ) f ( Z ε [ u ] ε , . . . , Z ε [ u n ] ε ) . Since [ σ M ( X ε , Y ε )] ε is an ( F εs n ) n -stopping time, we can apply the op-tional sampling theorem to conclude that the terms involving ζ εi andΣ εi − ¯Σ εi vanish. Again by Proposition 4.2, we have E h [ s ] ε ∧ [ σ M ( X ε ,Y ε )] ε X i =[ u ] ε ∧ [ σ M ( X ε ,Y ε )] ε +1 Q εi (cid:12)(cid:12) F ε [ u ] ε i ≤ δ ( ε ) → ε ↓ C = 2 C . In (4.6), letting M ↑ M with the dominated convergence theoremand letting η ↓ E (cid:2) ( β ∧ Λ s ) F (cid:3) ≤ E (cid:2) ( β ∧ Λ u ) F (cid:3) Since Λ s is bounded from below by Proposition 2.9 (iii), the monotoneconvergence theorem yields the conclusion by β ↑ ∞ . Indeed, we obtainthe integrability of Λ s by applying this argument with u = 0. (cid:3) Now we are in turn to complete the proof of Theorem 1.1. Proof of Theorem 1.1. In Theorem 4.3, we have proved the existenceof a coupling ( X, Y ) of g ( τ ′ ( s ))- and g ( τ ′′ ( s ))-Brownian motions withdeterministic initial data such that s L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is a super-martingale. Thus it suffices to show that we can choose the family oflaws of couplings as a measurable function of initial data.To complete it, we shall employ a measurable selection theorem.Note that the space of all Borel probability measures on the path space C ([0 , S ] → M × M ) equipped with the weak topology is a Polish space.We define K as the set of all laws of a coupling ( X, Y ) of g ( τ ′ ( s ))-Brownian motion X = ( X s ) ≤ s ≤ S and g ( τ ′′ ( s ))-Brownian motion Y =( Y s ) ≤ s ≤ S such that(a) ( X , Y ) is deterministic and(b) For 0 ≤ u ≤ s , it holds that L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is integrable and E [ L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) |F u ] ≤ L τ ′ ( u ) ,τ ′′ ( u )0 ( X u , Y u ) a.s.We denote the probability or the expectation with respect to Q ∈ K by P Q or E Q respectively. For each m ′ , m ′′ ∈ M , let us define K m ′ ,m ′′ ⊂K by K m ′ ,m ′′ := { Q ∈ K | P Q [( X , Y ) = ( m ′ , m ′′ )] = 1 } . By [3, Theorem 6.9.6], the claim holds once we show that K m ′ ,m ′′ is com-pact for each m ′ , m ′′ . Since the marginal distributions of elements in K m ′ ,m ′′ is fixed, the Prokhorov theorem yields that K m ′ ,m ′′ is relativelycompact. To show that K m ′ ,m ′′ is closed, take a sequence Q n ∈ K m ′ ,m ′′ which converges to Q . The following argument is similar to the one inthe proof of Theorem 4.3. First, P Q [( X , Y ) = ( m ′ , m ′′ )] = 1 obviouslyholds and hence Q verifies the condition (a). Second, for each k ∈ N , f i ∈ C b ( M ) ( i = 1 , . . . k ), 0 ≤ u ≤ u ≤ · · · ≤ u k ≤ s ≤ S and R > E Q n " L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) ∧ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 f i ( X u i , Y u i ) ≤ E Q n " L τ ′ ( u k ) ,τ ′′ ( u k )0 ( X u k , Y u k ) ∧ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 f i ( X u i , Y u i ) . COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 35 Thus, by tending n → ∞ and R → ∞ after it, we obtain E Q " L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 f i ( X u i , Y u i ) ≤ E Q " L τ ′ ( u k ) ,τ ′′ ( u k )0 ( X u k , Y u k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k Y i =1 f i ( X u i , Y u i ) . In particular, by applying the same argument for k = 1, f ≡ u = 0, we obtain that L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is integrable with respect to Q . Thus Q verifies the condition (b) and hence Q ∈ K m ′ ,m ′′ . It means K m ′ ,m ′′ is closed and the proof is completed. (cid:3) Proof. of Corollary 1.2. Since Proposition 2.9 (iii) and Lemma 2.12ensure that L t ′ ,t ′′ is continuous and bounded from below, there is aminimizer π ∈ Π (cid:0) c ′ ( t ′ ) , c ′′ ( t ′′ ) (cid:1) for C t ′ ,t ′′ ,ϕ (cid:0) c ′ ( t ′ ) , c ′′ ( t ′′ ) (cid:1) = C τ ′ (0) ,τ ′′ (0)0 ,ϕ (cid:0) c ′ ( τ ′ (0)) , c ′′ ( τ ′′ (0)) (cid:1) . Let P ( m ′ ,m ′′ ) be the law of the coupling ( X, Y ) with ( X , Y ) = ( m ′ , m ′′ )obtained in Theorem 1.1. This is a probability measure on W ( M ) × W ( M ), where W ( M ) is the space of continuous paths in M defined on[0 , S ]. Let P be the probability measure on W ( M ) × W ( M ) given by P (d w ′ , d w ′′ ) := Z M × M π (d m ′ , d m ′′ ) P ( m ′ ,m ′′ ) (d w ′ , d w ′′ ) . Note that P is well-defined by virtue of the measurability result in The-orem 1.1. Under P , the canonical process ( w ′ ( s ) , w ′′ ( s )) is a coupling of g ( τ ′ ( s ))-Brownian motion and g ( τ ′′ ( s ))-Brownian motion with the ini-tial distribution π such that L τ ′ ( s ) ,τ ′′ ( s )0 ( w ′ ( s ) , w ′′ ( s )) is a supermartin-gale. In particular, the law of ( w ′ ( s ) , w ′′ ( s )) gives a coupling of c ′ ( τ ′ ( s ))and c ′′ ( τ ′′ ( s )) for each s ∈ [0 , S ]. Since ϕ (cid:0) L τ ′ ( s ) ,τ ′′ ( s )0 ( w ′ ( s ) , w ′′ ( s )) (cid:1) isstill a supermartingale under P , we have C τ ′ ( s ) ,τ ′′ ( s )0 ,ϕ (cid:16) c ′ ( τ ′ ( s )) , c ′′ ( τ ′′ ( s )) (cid:17) ≤ Z M × M E ( m ′ ,m ′′ ) h ϕ (cid:16) L τ ′ ( s ) ,τ ′′ ( s )0 ( w ′ ( s ) , w ′′ ( s )) (cid:17) i π (d m ′ , d m ′′ ) ≤ Z M × M E ( m ′ ,m ′′ ) h ϕ (cid:16) L τ ′ (0) ,τ ′′ (0)0 ( w ′ (0) , w ′′ (0)) (cid:17) i π (d m ′ , d m ′′ )= Z M × M ϕ (cid:16) L τ ′ (0) ,τ ′′ (0)0 ( m ′ , m ′′ ) (cid:17) π (d m ′ , d m ′′ )= C τ ′ (0) ,τ ′′ (0)0 ,ϕ (cid:16) c ′ ( τ ′ (0)) , c ′′ ( τ ′′ (0)) (cid:17) , where E ( m ′ ,m ′′ ) stands for the expectation with respect to P ( m ′ ,m ′′ ) .Hence C t ′ − s,t ′′ − s ,ϕ (cid:0) c ′ ( t ′ − s ) , c ′′ ( t ′′ − s ) (cid:1) is non-increasing in s because we can repeat the same argument even if we replace the initial time 0with any s ′ ∈ [0 , s ]. (cid:3) Remark 4.2. The proof of the integrability of the L -distance betweenthe coupling of Brownian motions by space-time parallel transport in[15] seems to be incorrect. It is worth mentioning that we can recoverthe same integrability as in Theorem 4.3 even in that case by the sameargument.More precisely, the argument in [15, Lemma 6] seems to require somemodification. In the proof, they claimed an inequality analogous toDoob’s L p -martingale inequality for a positive supermartingale. How-ever it is not true in general. Indeed there is a counterexample toDoob’s inequality when p = 1 for a positive martingale M n (See [8,Example 5.4.2]). Since the x x /p is nonincreasing and concave on[0 , ∞ ) for p ≥ M /pn gives a counterexample to Doob’s L p -inequalityto positive supermartingales.On the other hand, if we further assume the stronger restriction onthe Ricci curvature, we can recover a similar integrability as stated in[15, Lemma 6]: Proposition 4.4. Suppose sup ≤ t ≤ T | Ric | g ( t ) < ∞ . Fix arbitrary twopoints m ′ , m ′′ ∈ M . For any coupling ( X s , Y s ) ≤ s ≤ S of g ( τ ′ ( s )) -Brownianmotion X = ( X s ) ≤ s ≤ S with X = m ′ and g ( τ ′′ ( s )) -Brownian motion Y = ( Y s ) ≤ s ≤ S with Y = m ′′ , sup ≤ s ≤ S L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) is integrable.Proof. Let o ∈ M be a fixed reference point of M . By Proposition2.9(iii) and Proposition 2.10(i), we have estimatesconst . ≤ L τ ′ ( s ) ,τ ′′ ( s )0 ( X s , Y s ) ≤ const . n ρ g ( τ ′ ( s )) ( X s , o ) + ρ g ( τ ′′ ( s )) ( o, Y s ) o + const . where the constants depend on sup ≤ t ≤ T | Ric | g ( t ) , T , t ′ and t ′′ but noton s . Therefore, for our statement, it is sufficient to prove the integra-bility of sup ≤ s ≤ T ρ g ( τ ′ ( s )) ( X s , o ) . By [14, Theorem 2], we see thatd ρ ( s, X s ) = n 12 ∆ g ( τ ′ ( s )) − ∂∂s o ρ ( s, X s )d s + ( U s .e i ) ρ ( s, X s )d W is − d L s , denoting ρ ( s, x ) := ρ g ( τ ′ ( s )) ( o, x ), where L s is a nondecreasing contin-uous process which increases only when X s belongs to the cut locusof o with respect to g ( τ ′ ( s )). Note that the set of s ∈ [0 , S ] where X s is in g ( τ ′ ( s ))-cut locus has null Lebesgue measure and hence other COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 37 quantities are also well-defined. Therefore we haved ρ ( s, X s ) = n ρ ( s, X s ) (cid:16) ∆ g ( τ ′ ( s )) − ∂∂s (cid:17) ρ ( s, X s )d s + d X i =1 | ( U s .e i ) ρ ( s, X s ) | o d s − ρ ( s, X s )d L s + 2 ρ ( s, X s )d β s where d β s = ( U s .e i ) ρ ( s, X s )d W is is a one-dimensional Brownian motion.Furthermore, Proposition 2 in [14] shows that ρ ( s, X s ) n 12 ∆ g ( τ ′ ( s )) − ∂∂s o ρ ( s, X s ) ≤ d − ρ ( s, X s ) n k coth( k · ρ ( s, X s ) ∧ r ) + k · ρ ( s, X s ) ∧ r o for some positive constants k and r from which we find that there issome constant c > ρ ( s, X s ) n 12 ∆ g ( τ ′ ( s )) + ∂∂s o ρ ( s, X s ) ≤ cρ ( s, X s ) for each s ≥ d X i =1 | ( U s .e i ) ρ ( s, X s ) | = 1 . So we shall apply a com-parison argument between ρ ’s SDE and (cid:26) d Z s = √ Z s ∨ β s + ( cZ s + 1)d s,Z = ρ (0 , X ) = ρ g ( t ′ ) ( m ′ , o ) . It is well-known that there is a global unique strong solution Z =( Z s ) s ≥ to the above SDE and this satisfies Z s ≥ s ≥ ρ ( s, X s ) ≤ Z s for each s ≥ ≤ u ≤ s Z pu is integrablefor any p ≥ (cid:3) References [1] Arnaudon, M.; Coulibaly, K. A. and Thalmaier, A.: Horizontal Diffusion in C Path Space. S´eminaire de Probabilit´es XLIII , Springer, 2010, 73–94.[2] Arnaudon, M.; Coulibaly, K. A. and Thalmaier, A.: Brownian motion withrespect to a metric depending on time: definition, existence and applications toRicci flow. C. R. Math. Acad. Sci. Paris (2008), no. 13-14, 773–778.[3] Bogachev, V.: Measure theory. Vol. I,II Springer-Verlag, Berlin, 2007.[4] Chen, B. and Zhu, X.: Uniqueness of the Ricci flow on complete noncompactmanifolds. J. Differential Geom. (1) (2006), 119–154.[5] Cheng, L.: The radial part of Brownian motion with respect to L -distance underRicci flow. arXiv:1211.3626v2 [6] Chow, B.; Chu, S.-C.; Glickenstein, D.; Guenther, C.; Isenberg, J.; Ivey, T.;Knopf, D.; Lu, P.; Luo, F. and Ni, L.: The Ricci flow: techniques and applica-tions. Part I. Geometric aspects. Mathematical Surveys and Monographs, .American Mathematical Society, Providence, RI, 2007. [7] Cranston, M.: Gradient estimates on manifolds using coupling. J. Funct. Anal. (1991), no. 1, 110–124.[8] Durrett, R.: Probability: theory and examples. Fourth edition Cambridge Se-ries in Statistical and Probabilistic Mathematics, Cambridge University Press,Cambridge, 2010.[9] Hamilton, R. S.: Three-manifolds with positive Ricci curvature. J. DifferentialGeom. (1982), no. 2, 255–306.[10] Ikeda, N.; Watanabe, S. Stochastic differential equations and diffusion pro-cesses. Second edition. North-Holland Mathematical Library, . North-HollandPublishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.[11] Kendall, W. S.: Nonnegative Ricci curvature and the Brownian coupling prop-erty. Stochastics (1986), no. 1-2, 111–129.[12] Kendall, W. S.: From stochastic parallel transport to harmonic maps Newdirections in Dirichlet forms, Amer. Math. Soc., Providence, RI; InternationalPress, Cambridge, MA, 1998, 49–115.[13] Kleiner, B. and Lott, J.: Notes on Perelman’s papers. Geom. Topol., ,(2008), no. 5, 2587–2855.[14] Kuwada, K. and Philipowski, R.: Non-explosion of diffusion processes on man-ifolds with time-dependent metric. Math. Z. (2011), no. 3-4, 979–991.[15] Kuwada, K. and Philipowski, R.: Coupling of Brownian motions and Perel-man’s L -functional J. Funct. Anal. (2011), 2742–2766.[16] Kuwada, K.: Convergence of time-inhomogeneous geodesic random walks andits application to coupling methods. Ann. Probab. (2012), no. 5, 1945–1979.[17] Lindvall, T. and Rogers, L. C. G.: Coupling of multidimensional diffusions byreflection. Ann. Probab. (1986), no. 3, 860–872.[18] Lott, J. Optimal transport and Perelman’s reduced volume. Calc. Var. PartialDifferential Equations (2009), no. 1, 49–84.[19] Mather, J. N. Action minimizing invariant measures for positive definite La-grangian systems. Math. Z. 207 (1991), no. 2, 169–207.[20] McCann, R. J. and Topping, P.: Ricci flow, entropy and optimal transporta-tion. Amer. J. Math. (2010), no. 3, 711–730.[21] Perel’man, G.: The entropy formula for the Ricci flow and its geometric ap-plications. arXiv preprint math/0211159 , 2002.[22] von Renesse, M.-K. and Sturm, K.-T.: Transport inequalities, gradient esti-mates, entropy and Ricci curvature. Comm. Pure. Appl. Math. (2005), no.7,923–940.[23] Shi, W.-X.: Deforming the metric on complete Riemannian manifolds. J. Dif-ferential Geom. (1) (1989), 223–301.[24] Topping, P.: Lectures on the Ricci flow. London Mathematical Society LectureNote Series, . Cambridge University Press, Cambridge, 2006.[25] Topping, P.: L -optimal transportation for Ricci flow. J. Reine Angew. Math. (2009), 93–122.[26] Villani, C.: Optimal transport. Old and new. Grundlehren der MathematischenWissenschaften, . Springer-Verlag, Berlin, 2009.[27] Wang, F.-Y.: Functional inequalities, Markov semigroups, and spectral theory. Science Press, Beijing, China, 2005. (T. Amaba) Ritsumeikan University, 1-1-1 Nojihigashi, Kusatsu, Shiga,525-8577, Japan E-mail address : [email protected] COUPLING OF BROWNIAN MOTIONS IN THE L -GEOMETRY 39 (K. Kuwada) Tokyo Institute of Technology, 2-12-1 ˆOokayama, Meguro-ku, Tokyo 152-8551, Japan E-mail address ::