Transport Inequalities, Gradient Estimates, Entropy and Ricci Curvature

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1 Transport Inequalities, Gradient Estimates, Entropy and Ricci Curvature Karl-Theodor Sturm & ax-konstantin von Renesse University of Bonn 1 Abstract. We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold in terms of convexity properties of the entropy (considered as a function on the space of probability measures on ) as well as in terms of transportation inequalities for volume measures, heat kernels and Brownian motions and in terms of gradient estimates for the heat semigroup. Keywords. transport inequalities, coupling, Wasserstein distance, displacement convexity, gradient estimate, entropy, Ricci curvature, Bakry-Emery estimate. 1 Introduction and statement of the main results For metric measure spaces there is neither a notion of Ricci curvature nor a common notion of bounds for the Ricci curvature (comparable for instance to Alexandrov s notion of bounds for the sectional curvature for metric spaces). The goal of this paper is to present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold in terms of convexity properties of the entropy (considered as a function on the space of probability measures on ) as well as in terms of transportation inequalities for volume measures, heat kernels and Brownian motions and in terms of gradient estimates for the heat semigroup. In the sequel, (, g) always is assumed to be a smooth connected Riemannian manifold with dimension n, Riemannian distance d(x, y) and Riemannian volume m(dx) = vol(dx). For r [1, [ the L r -Wasserstein distance of two measures µ 1, µ 2 on is defined as { d W r (µ 1, µ 2 ) := inf } 1/r d(x 1, x 2 ) r µ(dx 1 dx 2 ) : µ C(µ 1, µ 2 ) where C(µ 1, µ 2 ) denotes the set of all couplings of µ 1 and µ 2, that is, the set of all measures µ on with µ(a ) = µ 1 (A) and µ( A) = µ 2 (A) for all measurable A. Here and in the sequel measure on always means: measure on equipped with its Borel σ-field. The set of probability measures µ on with d(x, y)r µ(dy) < for some 1 Supported by the Collaborative Research Center SFB 611 of the German Research Council (DFG). 1

2 (hence all) x will be denoted by P r (). Equipped with the metric d W r, the space P r () is a geodesic space. The entropy is defined as a function on P r () by Ent(ν) := if ν is absolutely continuous w.r.t. vol with otherwise. dν dν log dx dx vol(dx) dν dx [ ] log dν dx vol(dx) < and Ent(ν) := + + Given an arbitrary geodesic space (X, ρ), a number K R and a function U : X ], + ] we say that U is K-convex iff for each (constant speed, as usual) geodesic γ : [, 1] X and for each t [, 1]: U(γ t ) (1 t) U(γ ) + t U(γ 1 ) K 2 t(1 t) ρ2 (γ, γ 1 ). K-convex functions on P 2 () are also called displacement K-convex (to make sure that t γ t is really the geodesic w.r.t. d W 2 and not the linear interpolation t (1 t)γ + tγ 1 in the space P 2 ()). Theorem 1. For any smooth connected Riemannian manifold and any K R the following properties are equivalent: (i) Ric() K, which always should be read as: Ric x (v, v) K v 2 for all x, v T x. (ii) The entropy Ent(.) is displacement K-convex on P 2 (). One reason for the importance of Theorem 1 is that it characterizes lower Ricci bounds referring neither to the differential structure of nor to the dimension of. Property (ii) may be formulated in any metric measure space. F. Otto & C. Villani [OV] gave a very nice heuristic argument for the implication (i) (ii). In the case K = this implication was proven in [CS1]. Remark 1. The argument in [OV] is based on a kind of formal Riemannian calculus on the space (P 2 (), d W 2 ). This calculus allows to interpret the heat flow Φ : (t, µ) µp t := µ(dx)pt (x,.) as the gradient flow w.r.t. the entropy Ent(.) on the space (P 2 (), d W 2 ). If the latter was a finite dimensional smooth Riemannian manifold then one easily could conclude that (ii) is equivalent to: (iii) The gradient flow Φ : R + P 2 () P 2 () w.r.t. Ent(.) satisfies d W 2 (Φ(t, µ), Φ(t, ν)) e Kt d W 2 (µ, ν) ( µ, ν P 2 (), t ). Indeed, both statements are equivalent. However, we will derive the equivalence of (ii) and (iii) only through their equivalence with (i). The equivalence of (i) and (iii) will be obtained as part of the more general Corollary 1 below. Here and henceforth, p t (x, y) always denotes the heat kernel on, i.e. the minimal positive fundamental solution to the heat equation ( t )p t(x, y) =. It is smooth in (t, x, y), symmetric in (x, y) and satisfies p t(x, y)vol(dy) 1. Hence, it defines a subprobability measure p t (x, dy) := p t (x, y)vol(dy) as well as operators p t : C c () C () and p t : L 2 () L 2 () 2

3 which are all denoted by the same symbol. Given µ P r () and t > we define a new measure µp t P r () by µp t (A) = A p t(x, y)µ(dx)vol(dy). Brownian motion on is by definition the arkov process with generator 1 2. Thus its transition (sub-)probabilities are given by p t/2. If the Ricci curvature of the underlying manifold is bounded from below then all the p t (x,.) are probability measures. If the latter holds true we say that the heat kernel and the associated Brownian motion are conservative. It means that the Brownian motion has infinite lifetime. The heat kernel on a Riemannian manifold is a fundamental object for analysis, geometry and stochastics. any properties and precise estimates are known. In most of these results, lower bounds on the Ricci curvature of the underlying manifold play a crucial role. Often, the results for the heat kernel on Riemannian manifolds turned out to be a source of inspiration for results in entirely different frameworks or much greater generality e.g. (hypo-)elliptic operators, Hodge- Laplacians, Feller generators or generators of Dirichlet forms instead of the Laplace-Beltrami operator; path spaces, spaces of measures or fractals instead of finite dimensional manifolds. Our second main result deals with robust versions of gradient estimates. Theorem 2. For any smooth connected Riemannian manifold and any K R the following properties are equivalent: (i) Ric() K. (iv) For all f C c (), all x and all t > (v) For all f C c () and all t > (vi) For all bounded f C Lip () and all t > p t f (x) e Kt p t f (x). p t f e Kt f. Lip(p t f) e Kt Lip(f). The equivalence of (i) and (iv), perhaps, is one of the most famous general results which relate heat kernels with Ricci curvature. It is due to D. Bakry &. Emery [BE84], see also [ABC + ] and references therein. Property (iv) is successfully used in various applications as a replacement (or definition) of lower Ricci curvature bounds for symmetric arkov semigroups on general state spaces. Our result states that (iv) can be weakened in two respects: one can replace the pointwise estimate by an estimate between L -norms; one can drop the p t on the RHS. Besides being formally weaker than (iv) one other advantage of (v) is that it is an explicit statement on the smoothing effect of p t whereas (iv) is implicit (since p t appears on both sides). As an easy corollary to the equivalence of the statements (iv) and (v) one may deduce the well known fact that (iv) is equivalent to the assertion that for all f, x and t as above p t f (x) e Kt [ p t ( f 2 )(x) ] 1/2. Property (vi) may be considered as a replacement (or as one possible definition) for lower Ricci curvature bounds for arkov semigroups on metric spaces. For several non-classical examples (including nonlocal generators as well as infinite dimensional or singular finite dimensional 3

4 state spaces) we refer to [Stu3], [DR2] and [vren3]. This property turned out to be the key ingredient to prove Lipschitz continuity for harmonic maps between metric spaces in [Stu3]. According to the Kantorovich-Rubinstein duality, property (vi) is equivalent to a contraction property for the heat kernels in terms of the L 1 -Wasserstein distance d W 1. Actually, however, much more can be proven: one obtains contraction in d W r for each r [1, ] and for any initial data; one obtains pathwise contraction for Brownian trajectories. Corollary 1. For any smooth connected Riemannian manifold and any K R the following properties are equivalent: (i) Ric() K. (vii) For all x, y and all t > there exists r [1, ] with d W r (p t (x,.), p t (y,.)) e Kt d(x, y). (viii) For all r [1, ], all µ, ν P r () and all t > : d W r (µp t, νp t ) e Kt d W r (µ, ν). (ix) For all x 1, x 2 there exists a probability space (Ω, A, P) and two conservative Brownian motions (X 1 (t)) t and (X 2 (t)) t defined on it, with values in and starting in x 1 and x 2, respectively, such that for all t > E [d(x 1 (t), X 2 (t))] e Kt/2 d(x 1, x 2 ). (x) There exists a conservative arkov process (Ω, A, P x, X(t)) x,t with values in such that the coordinate processes (X 1 (t)) t and (X 2 (t)) t are Brownian motions on and such that for all x = (x 1, x 2 ) and all t > d(x 1 (t), X 2 (t)) e Kt/2 d(x 1, x 2 ) P x -a.s. Note that each of the statements (vii) and (viii) implicitly includes the conservativity of the heat kernel. Indeed, the finiteness of the Wasserstein distance implies that the measures under consideration must have the same total mass. Thus p t (x, ) is constant in x, hence also constant in t and therefore equal to 1. Our interpretation of these results is as follows: if we put mass distributions µ and ν on and if they spread out according to the heat equation then the lower bound for the Ricci curvature of controls how fast the distances between these distributions may expand (or have to decay) in time. Our third main result deals with transportation inequalities for uniform distributions on spheres and analogous inequalities for uniform distributions on balls. Here the lower Ricci bound is characterized as a control for the increase of the distances if we replace Dirac masses δ x and δ y by uniform distributions σ r,x and σ r,y on spheres around x and y, resp. or if we replace them by uniform distributions m r,x and m r,y on balls around x and y, resp. 4

5 Theorem 3. For any smooth connected compact Riemannian manifold and any K R the following properties are equivalent: (i) Ric() K. (xi) The normalized Riemannian uniform distribution on spheres satisfies the asymptotic estimate σ r,x (A) := Hn 1 (A B r (x)) H n 1, A B() ( B r (x)) d W 1 (σ r,x, σ r,y ) where the error term is uniform w.r.t. x, y. (xii) The normalized Riemannian uniform distribution on balls satisfies the asymptotic estimate ( 1 K ) 2n r2 + o(r 2 ) d(x, y) (1) m r,x (A) := m(a B r(x)), A B() m(b r (x)) d W 1 (m r,x, m r,y ) where the error term is uniform w.r.t. x, y. ( K ) 1 2(n + 2) r2 + o(r 2 ) d(x, y) (2) The advantage of this characterization of Ricci curvature is that it depends only on the basic, robust data: measure and metric. It does not require any heat kernel, any Laplacian or any Brownian motion. It might be used as a guideline in much more general situations. For instance, let (, d) be an arbitrary separable metric space equipped with a measure m on its Borel σ-field and assume that (2) holds true (with some numbers K R and n > ). Define an operator m r acting on bounded measurable functions by m r f(x) = f(y) m r,x(dy). Then by the Arzela-Ascoli theorem there exists a sequence (l j ) j N such that ( p t f := lim m ) lj j 2(n+2)t/l f j exists (as a uniform limit) for all bounded f C Lip () and it defines a arkov semigroup on satisfying Lip(p t f) e Kt Lip(f) (cf. proof of the implication (xi) (i) of the above Theorem and proof of Theorem 4.3 in [Stu3]). 2 Ricci curvature and entropy Proof of Theorem 1. (ii) (i): Assume (i). Then Ric (e 1, e 1 ) K ɛ for some o, some unit vector e 1 T o and some ɛ >. Let e 1, e 2,..., e n be a ONB of T o such that R(e 1, e i )e 1 = k i e i 5

6 for suitable numbers k i, i = 1,..., n (denoting the sectional curvature of the plane spanned by e 1 and e i if i 1). Then n i=1 k i = Ric (e 1, e 1 ) K ɛ. For δ, r > let A 1 := B δ (exp o (re 1 )) and A := B δ (exp o ( re 1 )) be geodesic balls and let }) n A 1/2 := exp o ({y T o : (y i /δ i ) 2 1 with δ i := δ (1 + r 2 (k i + ɛ 2n )/2). Choosing δ r 1 we can achieve that γ 1/2 A 1/2 for each minimizing geodesic γ : [, 1] with γ A, γ 1 A 1. Now let µ and µ 1 be the normalized uniform distribution in A and A 1, resp. and let ν be the normalized uniform distribution in A 1/2. Then with c n := V ol(b 1 ) in R n whereas i=1 Ent(µ ) = log vola = log c n n log δ + O(δ 2 ) Ent(µ 1 ) = log vola = log c n n log δ + O(δ 2 ) Ent(ν) = log vola 1/2 = log c n = log c n n log δ r 2 (ɛ/2 + n log δ i + O(δ 2 ) i=1 n k i )/2 + O(r 4 ) + O(δ 2 ) i=1 log c n n log δ r 2 (K ɛ/2)/2 + O(r 4 ) + O(δ 2 ) Since the optimal mass transport from µ to µ 1 (w.r.t. d W 2 of µ 1/2 must be contained in the set A 1/2. Hence, ) is along geodesics of, the port Ent(µ 1/2 ) Ent(ν) and thus for δ r 1. Ent(µ 1/2 ) 1 2 Ent(µ ) 1 2 Ent(µ 1) K 2 r2 + ɛ 4 r2 + O(r 4 ) + O(δ 2 ) > K 8 dw 2 (µ, µ 1 ) 2 (i) (ii): Here we follow closely the argumentation of [CS1] and use their notation. Assume that Ric() K. We have to prove that Ent(µ t ) (1 t)ent(µ ) + tent(µ 1 ) K 2 t(1 t)dw 2 (µ, µ 1 ) 2 for each geodesic t µ t in (P 2 (), d W 2 ) and each t [, 1]. Without restriction, we may assume that µ and µ 1 are absolutely continuous (otherwise the RHS is infinite). Hence, there exists a unique geodesic connecting them. It is given as µ t = (F t ) µ where F t (x) = exp ( x ( t ϕ(x)) ( with a suitable ) function ϕ. oreover, with J t (x) ( := det df t (x) ( and s(r) ) := K sin n 1 )/ r K n 1 r K (which should be read as s(r) := sinh n 1 )/ r K n 1 r if 6

7 K < and as s(r) = 1 if K = ) and with v t (x, y) being the volume distortion coefficient of [CS1] we deduce Ent(µ t ) = Ent(µ ) log J t (x)µ (dx) and thus Ent(µ t ) + (1 t)ent(µ ) + tent(µ 1 ) = log J t (x)µ (dx) t log J 1 (x)µ (dx) [ n log (1 t)v 1 t (F 1 (x), x) 1/n + tv t (x, F 1 (x)) 1/n J 1 (x) 1/n] µ (dx) t log J 1 (x)µ (dx) [ [ ] s((1 t)d(f1 (x), x)) 1 1/n [ ] ] s(td(x, F1 (x))) 1 1/n n log (1 t) + t J 1 (x) 1/n µ (dx) s(d(f 1 (x), x)) s(d(f 1 (x), x)) t log J 1 (x)µ (dx) (n 1) [(1 t) log s((1 t)d(f 1 (x), x)) + t log s(td(f 1 (x), x)) log s(d(f 1 (x), x))] µ (dx) K 2 t(1 t) d 2 (F 1 (x), x)µ (dx) = K 2 t(1 t)dw 2 (µ, µ 1 ). Here the first and second inequality follow from [CS1], equations (73) and (24). The third inequality follows from the concavity of the logarithm and the last one from the fact that (1 t) log s ((1 t)r) + t log s (tr) log s(r) t(1 t) K 2 n 1 r2 = (1 t)λ((1 t)r) + tλ(tr) λ(r) for all r under consideration and t [, 1] since λ (r) where λ(r) := log s(r) n 1 r2. Note that according to the Bonnet-yers theorem we may restrict ourselves to r with K n 1 r2 π 2. In order to verify that λ (r) it suffices to consider the cases K = ±(n 1). If K = (n 1) then λ(r) = log sinh r log r 1 6 r2 and λ (r) = cosh r sinh r 1 r 1 3r. The latter is nonpositive for all r > if and only if r cosh r sinh r 1 3 r2 sinh r for all r >. Differentiating and dividing by r/3 we see that this is equivalent to r cosh r + sinh r which (again by differentiation) will follow from r sinh r which is obviously true. Analogously, if K = n 1 the condition λ (r) is equivalent to r cos r sin r r2 sin r which (by the same arguments as before) is equivalent to r sin r. Here of course, we have to restrict ourselves to r [, π]. K 7

8 3 Ricci curvature and transport inequalities for heat kernels and Brownian motions Proof of Theorem 2 and Corollary 1. (i) (iv): This is due to D. Bakry &. Emery [BE84] and can be obtained using their Γ 2 calculus (cf. [ABC + ], prop ). (iv) (v): Take. on both sides and use (on the RHS) the fact that p t is a contraction on L (). (v) (i): We prove it by contradiction, assuming (i) (v). If (i) is not true then there exists a point and v S n 1 T such that Ric (v, v) K ɛ for some ɛ >. Let F = {x \ Cut() log x v} be the orthogonal hypersurface to v in and define the signed distance function d ± F from F by d ± F : \ Cut() R, d± F (x) := dist(x, F)sign v, log x. (3) It is shown in Lemma 1 below that d ± F C (U) for some neighborhood U and that furthermore d ± F () = v (4) d ± F (x) = 1 x U (5) Hess (d ± F ) =. (6) Hence for any neighborhood U with U U and any smooth cut-off function ψ Cc (U) with ψ U 1 the function f = d ± F ψ inherits the properties (4), (5) and (6) with U replaced by U in (5). In particular, by continuity of the function Γ 2 (f, f) : U R Γ 2 (f, f)(x) := Hess x (f) 2 + Ric x ( f, f) we find Γ 2 (f, f) K 1 2 ɛ on some neighborhood of which contains w.l.o.g. U. If, moreover, (v) is true then for any nonnegative test function ϕ Cc and t 1 we obtain p t f 2 ϕ dvol p t f 2 ϕ dvol exp( 2Kt) ϕ dvol = (1 2Kt + o(t)) ϕ dvol. (7) We choose a test function ϕ Cc (U ) and define for t > Φ(t) := p t f 2 ϕ dvol. 8

9 Hence, since p t f 2 f 2 1 on p(ϕ) U, the function Φ extends continuously on the entire nonnegative half line by Φ() := ϕ. From Bochner s formula we deduce Φ (t) = 2 p t f, p t f ϕ dvol = = t = 2 (2 p t f 2 2Γ 2 (p t f, p t f))ϕ dvol 2 p t f 2 ϕ 2Γ 2 (p t f, p t f)ϕ dvol (2 f 2 ϕ 2Γ 2 (f, f)ϕ) dvol Γ 2 (f, f)ϕ dvol (ɛ 2K) ϕ = (ɛ 2K)Φ(). Thus t Φ(t) is differentiable in t = + with Φ (+) > (ɛ 2K)Φ(). Consequently we find for small t that Φ(t) Φ() + (ɛ 2K)Φ()t + o(t) = Φ()(1 2Kt + o(t)), i.e. p t f 2 ϕ dvol (1 + (ɛ 2K)t + o(t)) ϕ dvol which is a contradiction to (7). (vi) (v): Trivial (vii) (vi): By Hölder s inequality, property (vii) for some r 1 implies property (vii) for r = 1 which in turn implies (vi) according to the Kantorovich-Rubinstein duality. Or explicitly: for each coupling λ of p t (x,.) and p t (y,.) p t f(x) p t f(y) = [f(z) f(w)]λ(dzdw) Lip(f) d(z, w)λ(dzdw) Hence, [ Lip(f) d(z, w) r λ(dzdw)] 1/r. p t f(x) p t f(y) Lip(f) d W r (p t (x,.), p t (y,.)) Lip(f) d(x, y) e Kt. (viii) (vii): Choose µ = δ x, ν = δ y. (ix) (vii): The distribution λ(.) := P ((X 1 (2t), X 2 (2t)).) of the pair (X 1 (2t), X 2 (2t)) defines a coupling of p t (x 1,.) and p t (x 2,.). Hence, d W 1 (p t (x 1,.), p t (x 2,.)) d(z 1, z 2 ) λ(dz 1 dz 2 ) = E [d(x 1 (2t), X 2 (2t))] e Kt d(x 1, x 2 ). 9

10 (x) (viii): Let λ be an optimal coupling of µ and ν w.r.t. d W r and let Π t be the transition semigroup of the arkov process from (x). Then λ t := λπ 2t is a coupling of µp t and νp t. Hence, d W r (µp t, νp t ) r d(w 1, w 2 ) r λ t (dw 1 dw 2 ) = d(w 1, w 2 ) r Π 2t ((x 1, x 2 ), dw 1 dw 2 )λ(dx 1 dx 2 ) = E (x 1,x 2 ) [d(x 1 (2t), X 2 (2t)) r ] λ(dx 1 dx 2 ) e Ktr d(x 1, x 2 ) r λ(dx 1, dx 2 ) (x) (ix): Take expectations. = e Ktr d W r (µ, ν) r. (i) (x) is well known and can be shown using either SDE theory on Riemannian manifolds in order to construct the coupling by parallel transport process on for two Brownian motions (cf. [Wan97] and references therein) or by a central limit theorem for coupled geodesic random walks and an estimate of the type (2) (cf. [vren3] for a similar argument). Remark 2. There are two more very natural candidates which one could consider as a test function in the proof of the (v) (i) part above, namely f 1 (z) := log (z), v T and f 2 (z) = d(exp (Lv), z) with L being large compared to U. However, even if f 1 () = f 2 () = v none of the two will help because f 1 1 in a neighborhood of and Hess f 2, which were essential properties of d ± F in our proof. Lemma 1. Let be a Riemannian manifold,, v T and F = {exp (u) u T, u v} the (n 1)-dimensional hypersurface through orthogonal to v. Then the signed distance function d ± F : R belongs to C (U) for some neighborhood U and Hess (d ± F ) =. Proof. The level sets F ɛ = {x d ± F (x) = ɛ} of d± F for ɛ 1 define a foliation of (a sufficiently small) neighborhood U by smooth hypersurfaces. The unit normal vector field to F ɛ is given by ν = d ± F which is well defined and smooth sufficiently close to F (d± F is a distance function on U in the sense of [Pet98], sec. 2.3.). Hence the Hessian of d ± F in a point p U may be interpreted as the shape operator of F ɛ in p F ɛ with ɛ = d ± F (p), i.e. Hess p (X, X) = Π Fɛ p (X, X) = S Fɛ (X), X Tp, p where Π Fɛ p is the second fundamental form of the hypersurface F ɛ and S Fɛ : T p T p F ɛ is the associated shape operator. The claim Hess (d ± F ) = then follows from the construction of F which implies that F = F is flat in, i.e. S F =. 1

11 4 Ricci curvature and transport inequalities for uniform distributions on spheres and balls Proof of Theorem 3. (xi) (i): Let us define the family of arkov operators σ r : F B F B on the set F B of bounded Borel measurable functions on by σ r f(x) = f(y)σ r,x(dy). Using that for f C 3 () σ r f(x) = f(x) + r2 2n f(x) + o(r2 ) (8) and an appropriate version of the Trotter-Chernov product formula (cf. thm in [EK86], applied to p t = exp(t ) as Feller semigroup on (C(),. )) we find for all f C() ( σ ) j j 2nt/j f(x) p t f(x) uniformly in x and locally uniformly in t. By the Rubinstein-Kantorovich duality condition (xi) implies ( σ r f(x) σ r f(y) 1 K ) 2n r2 + o(r 2 ) d(x, y) Lip(f) for all f C Lip () and x, y, i.e. Lip(σ r f) ( 1 K 2n r2 + o(r 2 ) ) Lip(f) and hence by iteration for j N, r = 2nt/j ( ( ) Lip σ j 2nt/j) f ( 1 Kt/j + o(t/j) ) j Lip(f). Passing to the limit for j yields which is equivalent to (i) by Theorem 2. Lip(p t f) exp( Kt) Lip(f), For the proof of the converse we construct an explicit transport from σ r,x to σ r,y in the following lemma, whose proof is given below. Lemma 2. Let be a smooth connected compact Riemannian manifold and for x let σ r,x (.) denote the normalized Riemannian uniform distribution on S r (x) := B r (x). Then for r sufficiently small for each x, y there exits a geodesic segment γ = γ xy and a measurable map Ψ x,y r : S r (x) S r (y) such that the push forward measure Ψ x,y r, σ r,x equals σ r,y and z S r(x) y log x z / 1 γ where the error term o(r 2 ) is uniform in x. d(x, y) log y Ψ x,y r (z) = o(r 2 ) (9) 11

12 (i) (xi): We show it for the case K <, the case K is treated analogously. Due to Lemma 2 we can assume that up to a negligible error Ψ x,y r is indeed nothing but parallel transport because d(z, Ψ x,y r (z)) d(z, exp y ( / γxy log x (z))) + d(exp y ( / γ log x (z)), Ψ x,y r (z)) d(z, exp y ( / γ log x (z))) + L / γ log x (z) log y Ψ x,y r (z) d(z, exp y ( / γ log x (z))) + Ld(x, y)o(r 2 ), where L is some uniform upper bound for the Lipschitz constant of log. (.) with respect to the second argument. The asymptotic inequality (1) is now easily verified from (8), since d W 1 1 (σ r,x, σ r,y ) H n 1 d(z, Ψ x,y r (z))h n 1 (dz) ( B r (x)) 1 = H n 1 ( B r (x)) B r(x) B r(x) = d(x, y) + r2 2n Dx,y (x) + d(x, y)o(r 2 ) with z D x,y (z) = d(z, exp y / γ log x z). Since d(z, exp y / γ log x z)h n 1 (dz) + d(x, y)o(r 2 ) D x,y (x) = trace Hess x D x,y = n I γ (J i, J i ) i=2 where I γ (J i, J i ) is the Indexform of along γ xy applied to the Jacobi field induced from parallel geodesic varations of γ in the direction e i with { γ e xy, e 2,, e n } being an orthonormal basis of T x. Hence we may conclude by the standard Ricci comparison argument that K D x,y (x) 2(n 1) n 1 such that we finally arrive at cosh ( K n 1 d(x, y)) 1 sinh ( K n 1 d(x, y)) Kd(x, y) d W 1 (σ r,x, σ r,y ) d(x, y) r2 2n Kd(x, y) + d(x, y)o(r2 ). (xii) (i): This is shown in the same way as the implication (xi) (i) with the slight difference that instead of (8) one uses m r f(x) = f(x) + r 2 2(n + 2) f(x) + o(r2 ). (i) (xii): We proceed as before for (i) (xi) where now we have to construct a map Φ x,y r which preserves the normalized uniform distributions on balls. However, since similarly to the condition (16) in the proof of Lemma 2 we have m r,x (A) = 1 r σ u,x (A)H n 1 ( B u (x))du m(b r (x)) 12

13 such a map can be constructed from a map Ψ x,y r 1,r 2 : B r1 (x) B r2 (y) with (Ψ x,y r 1,r 2 ) σ r1,x = σ r2,y and which is almost induced from parallel transport in the sense of (9). It is clear that Lemma 2 can easily be generalized to yield such a map Ψ x,y r 1,r 2 which is all we need. Proof of Lemma 2. We show the lemma for the two dimensional case first and inductively generalize this result to higher dimensions later. Let n = 2 and choose a parametrization of S r (x) and S r (y) (using Riemannian polar coordinates, for example) on S 1 R 2 T x, i.e. for all f : S r (x) R S r(x) f(z)h n 1 (dz) = with a density D x (r, ϑ) given by D x (t, ϑ) = S 1 D x (r, ϑ) f(ϑ)s(dϑ) = 2π det ( Y i (t, ϑ), Y j (t, ϑ) ) ij D x (r, s) f(s)ds = t n 1( 1 t2 6 c x(ϑ, ϑ) + o(t 2 ) ), (1) where c x is the Gaussian curvature of (, g) in x and Y j (t, ϑ) is the Jacobi field along t exp x (tϑ) with J i () = and J i () = e i for an orthonormal basis {e i i = 1, 2} of T x (cf. [GHL9], for instance). For x, y and γ xy fixed let the parametrization of S 1 x T x and S 1 y T y on [, 2π] be chosen in such a way that Next, we choose a function τ = τ x,y r 1 H n 1 (S r (x)) S 1 x γ xy () und S 1 y γ xy ((d(xy)). u : [, 2π] [, 2π] with τ() = satisfying D x (r, s)ds = 1 H n 1 (S r (y)) τ(u) D y (r, s)ds (11) for all u [, 2π]. Identifying τ : [, 2π] [, 2π] with the associated τ : S 1 S 1 then (11) just means that the induced map Ψ = Ψ x,y : S r (x) S r (y) transports the measure H n 1 r,x Denoting Ψ(z) := exp y (rτ( 1 r log x(z))) into Hr,y n 1. By definition of Ψ x,y estimate (9) is equivalent to z [,2π] y τr x,y (z) z = o(r). (12) d(x, y) E x (r, s) := D x(r, s) H n 1 (S r (x)) = 1 + O(r2 ), (13) with E. (., s) C 2 ( [, ɛ]) for all s [, 2π] and some ɛ >, (11) yields τ(z) (1 + O(r 2 z )) τ(z) z = E y (r, s)ds = (E x (r, s) E y (r, s)) ds. z 13

14 Consequently y τr x,y (z) z z (1 + O(r 2 )) x E. (r, s) ds. d(x, y) Due to E. (, s) 1 = const we find lim r x E. (r, s) = and hence lim r y τr x,y (z) z z lim rd(x, y) r = lim r = lim r z z r xe. (r, s) ds xe. (r, s) x E. (r, s), r xe. (r, s) Txds xe. (r, s) x E. (r, s), x r E.(r, s) Txds. (14) The right hand side of (13) yields r r= E.(r, s) = from which we see that the integral above vanishes for r tending to. This establishes (12) for fixed x. By the smoothness of (, g) the error term is locally uniform in x and hence it is also globally uniform since is compact. The case dim = 3 will show how we can deal with arbitrary dimensions n N. Fix x, y as well as some segment γ xy from x to y. By means of the inverse of the exponential map we lift the measures σ r,x and σ r,y onto the unit spheres in T x and T y respectively which we disintegrate along the γ xy -direction as follows. Choose an o.n. basis {e 1, e 2, e 3 } T x with e 1 = γ xy and {e 1, e 2, e 3 } = / γ {e 1, e 2, e 3 } T y and for u [ 1, 1] let Sr,x(u) = ( {exp x r(ue1, se 2, te 3 ) ) u 2 + s 2 + t 2 = 1} denote the orthogonal part of S r (x) w.r.t. γ at ue 1. Define a probability measure c r,x (u)(du) on [ 1, 1] e 1 R T x by c r,x (u)(du) = = 1 H n 1 (S r (x)) Hn 2( Sr,x(u) ) du 1 S D 2 x (r, ϑ)s 2 D x (r, (u, η))s 1 (dη)du (dϑ) S 1 1 u 2 with the Riemannian volume density ) D x (t, ϑ) = t (1 n 1 t2 6 Ric x(ϑ, ϑ) + o(t 2 ) (15) and c r,y (u) analogously. Let τ 1 : [ 1, 1]e 1 [ 1, 1]e 1 be the function defined by s c r,x (u)du = τ 1 (s) c r,y (u)du s [ 1, 1]. (16) For each s [ 1, 1] define a transport τ s : S r,x(s) S r,y(τ 1 (s)), 14

15 analogously to (11) which preserves the probability measures σ r,x(s)(.) and σ r,y(τ(s))(.) that are obtained from conditioning σ r,x and σ r,y on S r,x(s) and S r,y(τ 1 (s)) respectively. Hence the map Ψ x,y : S r (x) S r (y), Ψ x,y (z) := exp y (rτ( 1 r log x(z))) induced from τ = τr x,y : T x S 2 (x) S 2 (y) T y τ(u) := ( τ 1 (u 1 ), τu 1 ((u 2, u 3 )) ) will push forward σ r (x) into σ r (y) and it remains to prove the asymptotic estimate (12). Since the distance τr x,y (u) u 2 is Euclidean we may use the estimate (12) from the two dimensional case for the τr x,y ((u 2, u 3 )) (u 2, u 3 ) part which persists also in this situation. Indeed, it is sufficient to note that the expression (14) and hence the error estimate z [,2π] y τx,y((u 2, u 3 )) (u 2, u 3 ) d(x, y) = o(r) holds true also for the embedded orthogonal spheres S r,x(u), S r,y(u) since they are parallel translates of one another and that by triangle inequality this also generalizes to the situation τ x,y : S r 1 (x) S r 2 (y) with r 1, r 2 r. Thus it remains to prove u 1 [ 1,1] y τ 1 u 1 u 1 d(x, y) = o(r), which follows from (16) by similar arguments which established (12) in the 2D-case. This completes the proof in three dimensions. For arbitrary n N one proceeds in a similar fashion by inductively reducing the problem to lower dimensions. References [ABC + ] Cécile Ané, Sébastien Blachère, Djalil Chafaï, Pierre Fougères, Ivan Gentil, Florent alrieu, Cyril Roberto, and Grégory Scheffer. Sur les inégalités de Sobolev logarithmiques. Société athématique de France, Paris, 2. [BE84] [CS1] [DR2] [EK86] [GHL9] [OV] Dominique Bakry and ichel Emery. Hypercontractivite de semi-groupes de diffusion. C. R. Acad. Sci., Paris, Ser. I, 299: , Dario Cordero-Erausquin, Robert ccann, and ichael Schmuckenschläger. A Riemannian interpolation inequality à la Borell, Brascamb and Lieb. Invent. ath., 146: , 21. Giuseppe Da Prato and ichael Röckner. Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Relat. Fields, 124: , 22. Stewart N. Ethier and Thomas G. Kurtz. arkov processes Characterization and convergence. John Wiley & Sons Inc., New York, Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine. Riemannian geometry. Springer-Verlag, Berlin, second edition, 199. Felix Otto and Cedric Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173(2):361 4, 2. 15

16 [Pet98] Peter Petersen. Riemannian geometry. Springer-Verlag, New York, [Stu98] [Stu3] [Vil3] [vren3] Karl-Theodor Sturm. Diffusion processes and heat kernels on metric spaces. Ann. Probab., 26(1):1 55, Karl-Theodor Sturm. A semigroup approach to harmonic maps. Prepint no. 39, SFB 611 Univ. Bonn, 23. Cedric Villani. Topics in ass Transportation. Graduate Studies in athematics. American athematical Society, 23. ax-k. von Renesse. Intrinsic Coupling on Riemannian anifolds and Polyhedra. Prepint no. 64, SFB 611 Univ. Bonn, 23. [Wan97] Feng-Yu Wang. On estimation of the logarithmic Sobolev constant and gradient estimates of heat semigroups. Probab. Theory Related Fields, 18(1):87 11, Address: Institut für Angewandte athematik, Universität Bonn Wegelerstrasse 6, Bonn Fax: KTS: sturm@uni-bonn.de, KvR: kostja@iam.uni-bonn.de 16