Displacement convexity of generalized relative entropies. II

Transcription

1 Displacement convexity of generalized relative entropies. II Shin-ichi Ohta and Asuka Takatsu arch 3, 23 Abstract We introduce a class of generalized relative entropies inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci curvature and the convexity of another weight function used in the definition of the generalized relative entropies. This convexity condition corresponds to Lott and Villani s version of the curvature-dimension condition. As applications, we obtain appropriate variants of the Talagrand, HWI and logarithmic Sobolev inequalities, as well as the concentration of measures. We also investigate the gradient flow of our generalized relative entropy. SC 2): 53C2, 58J35 Primary); 28A33, 49Q2 Secondary) Keywords: Bregman divergence, Ricci curvature, optimal transport, curvaturedimension condition, gradient flow Contents Introduction 2 2 Preliminaries 6 2. Weighted Riemannian manifolds Wasserstein geometry Information geometry Information geometry continued: The case of ϕ m s) = s 2 m Displacement convexity classes DC N 3 4 Admissible spaces 6 Department of athematics, Kyoto University, Kyoto , Japan sohta@math.kyoto-u.ac.jp); Supported in part by the Grant-in-Aid for Young Scientists B) Graduate School of athematics, Nagoya University, Nagoya , Japan takatsu@math.nagoyau.ac.jp); Supported in part by the Grant-in-Aid for Young Scientists B)

2 5 ϕ-relative entropy and its displacement convexity Curvature-dimension condition ϕ-relative entropy H ϕ Displacement convexity of H ϕ Functional inequalities 3 7 Concentration of measures General estimate Concentration of measures mϕ)-normal concentration Gradient flow of H ϕ : Compact case Geometric structure of P), W 2 ) Gradient flows in P), W 2 ) H ϕ and the ϕ-heat equation Proof of Claim Gradient flow of H ϕ : Noncompact case Riemannian structure of P 2 ), W 2 ) Gradient flow of H ϕ Remarks on construction and contraction Finsler case 6. Finsler manifolds Weighted Ricci curvature and nonlinear Laplacian Displacement convexity of H ϕ and applications Gradient flow of H ϕ A Appendix: easure concentration via u ϕ -entropy inequality 67 Introduction This is a continuation of our work [OT] on the displacement convexity of generalized entropies and its applications. We consider more general entropies than [OT] and generalize most results in appropriate ways. Some of our observation shall shed new light on [OT]. It has been known since the celebrated work of ccann [c] that the convexity of an energy entropy) functional along geodesics in the Wasserstein space plays a vital role in the study of the existence and the uniqueness of a ground state a minimizer of the energy). Here the quadratic) Wasserstein space over a complete separable metric space X, d) is the space P 2 X) of Borel probability measures on X having finite second moments, endowed with the Wasserstein distance function W 2 derived from the onge Kantorovich mass transport problem see Subsection 2.2). We say that a functional S on P 2 X) is displacement K-convex for K R Hess S K for short) if any pair 2

3 µ, µ P 2 X) can be joined by a minimal geodesic µ t ) t [,] in P 2 X), W 2 ) such that Sµ t ) t)sµ ) + tsµ ) K 2 t)tw 2µ, µ ) 2 holds for all t [, ]. As usual, the displacement -convexity may be simply called the displacement convexity. The word displacement is inserted for avoiding a confusion with the convexity along the linear interpolation S t)µ +tµ ) t)sµ )+tsµ ). Since we deal with only the displacement convexity, we may sometimes omit displacement. As any geodesic in the Wasserstein space is written as the transport along geodesics in the underlying metric space, the displacement convexity of an energy functional can be derived from the convexity of its generating function. For instance, let Su Ψ be the free energy functional on P 2 R n ) consisting of the internal energy and the potential energy as ) dµ Su Ψ µ) := u dl n + Ψ dµ R dl n n R n for absolutely continuous probability measures µ on R n with respect to the Lebesgue measure L n, where the energy density u is a function on R and the potential Ψ is a function on R n. Then Su Ψ is strictly displacement convex if u is convex and satisfies certain additional conditions to be precise, see Definition 3.) and Ψ is strictly convex, and the unique ground state ν := σl n satisfies u σ) = Ψ+λ with a normalizing constant λ. We mention that the uniqueness is measured at the level of the energy functional, that is, we have Su Ψ µ) Su Ψ ν) and equality holds if and only if µ = ν. oreover, the displacement convexity of the free energy Su Ψ is a crucial tool also in the investigation of the asymptotic behavior of the solution to the associated evolution equation ρ t = div ρ [u ρ)] + ρ Ψ ) by regarding it as the gradient flow of Su Ψ in the Wasserstein space see [JKO], [AGS], [CV] and [CV2] among others). In particular, the heat flow is regarded as the gradient flow of the relative entropy with respect to the Lebesgue measure) ) dµ dµ Ent L nµ) := R dl ln dl n, n n dl n which is also called the Kullback Leibler divergence in information theory. On curved spaces such as Riemannian manifolds, the displacement convexity of energy functionals is related to the curvature of the underlying space, that is a crucial difference from the convexity along linear interpolations t)µ + tµ. On a Riemannian manifold equipped with the Riemannian volume measure vol g, the relative entropy is similarly defined by ) dµ dµ Ent volg µ) := ln dvol g. dvol g dvol g It has been shown by von Renesse and Sturm [vrs] inspired by [CS] and [OV]) that for any K R the following are mutually equivalent: 3

4 The relative entropy Ent volg is displacement K-convex on P 2 ), W 2 ). The Ricci curvature is bounded from below by K as Ric g v, v) K v, v for all v T. The heat flow is K-contractive in the sense that W 2 µ t, µ t ) e Kt W 2 µ, µ ) holds for all t and for any weak solutions ρ t ) t, ρ t ) t to the heat equation ρ/ t = ρ such that µ t := ρ t vol g, µ t := ρ t vol g P 2 ). See also [AGS], [AGS3], [Oh], [Sa] and [Vi2, Chapter 23] for the connection between the K-convexity of the functional and the K-contraction property of its gradient flow. The displacement K-convexity of Ent volg Hess Ent volg K) is called the curvaturedimension condition CDK, ) after Bakry and Émery s pioneering work [BE]. One remarkable point of CDK, ) is that it can be formulated on general metric measure spaces without any differentiable manifold) structure. Such metric measure spaces with Ricci curvature bounded below are independently investigated by Sturm [St2] and Lott and Villani [LV2], and known to enjoy several properties common to Riemannian manifolds of Ric g K. For example, as was indicated by Otto and Villani [OV], CDK, ) with K > implies various functional inequalities such as the Talagrand inequality, the HWI inequality, the logarithmic Sobolev inequality and the global Poincaré inequality [LV2, Section 6]). The curvature-dimension condition CDK, ) is generalized to CDK, N) for each K R and N, ]. On an n-dimensional complete connected Riemannian manifold, g) of n 2 equipped with a weighted measure ω = e f vol g with f C ), the condition CDK, N) is known to be equivalent to the lower bound of the N-Ricci curvature Ric N v, v) K v, v [St], [St3], [LV], [LV2], see Definition 2. for the definition of Ric N ). In particular, an unweighted Riemannian manifold, vol g ) satisfies CDK, N) if and only if its Ricci curvature is bounded below by K and its dimension is bounded above by N. We remark that Sturm s and Lott and Villani s definitions of the curvaturedimension condition are slightly different, though they are equivalent on non-branching spaces such as Riemannian or Finsler manifolds. In both cases it is a certain convexity condition of a class of entropies, and Lott and Villani s class is larger than Sturm s one. On non-branching metric measure spaces, the condition CD, N) for N [n, ) is equivalent to the displacement convexity of the Rényi entropy ) N )/N dµ S N µ) := dω. dω For K, however, CDK, N) is not simply the displacement K-convexity of S N. In fact, it was shown in [St] see also [OT, Theorem 4., Remark 4.32)] and [BS]) that, on a weighted Riemannian manifold, ω), Hess S N K can hold only for K and is equivalent to Ric N regardless of the value of K. It was also observed in [St, Theorem.7] for unweighted Riemannian manifolds that there are some functionals whose displacement K-convexity characterizes the combination of Ric K and dim N, whereas it is unclear if there are any applications of these entropies. In our previous work [OT], we introduced the m-relative entropy H m for the parameter m [n )/n, ), ) inspired by the Bregman divergence in information 4

5 theory/geometry see [Am], [AN]) as well as the Tsallis entropy in statistical mechanics see [Ts], [Ts2]). We fix a reference measure ν = exp m Ψ)ω on a weighted Riemannian manifold, ω) involving the m-exponential function exp m t) := max{ + m )t, } /m ), then the m-relative entropy of an absolutely continuous measure µ P 2 ) with respect to ν is given by up to an additive constant) { ) m dµ H m µ) := m dµ ) } m dν dω. mm ) dω dω dω This generalizes the relative and the Rényi entropies in the sense that lim m H m µ) = Ent ν µ) and that H m µ) = N{m S N µ) + } with N = / m) if Ψ i.e., ν = ω). Then the displacement K-convexity of H m is equivalent to the combination of Ric N of, ω)) and Hess Ψ K [OT, Theorem 4.]). We stress that N becomes negative for m >, then Ric N is defined in the same form as the case of N n, ) see Definition 2.). Similarly to CDK, ), we can derive from Hess H m K > the associated functional inequalities see also [AGK], [CGH], [Ta] for related works) and the concentration of measures in terms of exp m ). Furthermore, the gradient flow of H m produces weak solutions to the fast diffusion equation m < ) or the porous medium equation m > ) with drift of the form ρ t = div ω ) m ρm ) + ρ Ψ, where div ω is the divergence of, ω) see also [Ot], [Vi2, Theorem 23.9]). We remark that Sturm [St] studied a more general class of entropies on unweighted Riemannian manifolds, where Ric N = Ric for all N. Compared to it, [OT] gave a detailed investigation of a concrete class of entropies, on more general weighted Riemannian manifolds by choosing appropriate parameters N). In this article, we introduce the more general class of entropies, called the ϕ-relative entropies H ϕ, again inspired by information theory/geometry. Here ϕ :, ), ) is a non-decreasing, positive, continuous function. Roughly speaking, our new class corresponds to Lott and Villani s class of entropies in their definition of the curvaturedimension condition, while the m-relative entropies in [OT] correspond to Sturm s class. The definition of H ϕ see Definition 5.3 for details) involves ν = exp ϕ Ψ) with the ϕ-exponential function exp ϕ which is the inverse function of the ϕ-logarithmic function ln ϕ t) := t ϕs) ds. We recover exp m and H m from ϕs) = s 2 m. Our first main theorem Theorem 5.7) asserts that Hess H m K is equivalent to Hess H ϕ K for all ϕ s in a certain class. This actually corresponds to the equivalence between Sturm s and Lott and Villani s curvature-dimension conditions on weighted Riemannian manifolds. This reveals that H m is an extremal element among H ϕ s in the appropriate class, see [Ta2] for a related work. Similarly to H m, we can derive from Hess H ϕ K > the variants of the Talagrand, HWI, logarithmic Sobolev, and global 5

6 Poincaré inequalities Theorem 6.3) as well as the concentration of measures in terms of exp m for some m = mϕ) Theorem 7.9). oreover, the gradient flow of H ϕ in P 2 ), W 2 ) produces weak solutions to the ϕ-heat equation Theorems 8.7, 9.7) ρ t = div ω ρ ρ ϕρ) + ρ Ψ The article is organized as follows: We first review the basic notions of weighted Riemannian geometry, Wasserstein geometry and information geometry in Section 2. Then, after preparing necessary notions in Sections 3, 4, we define H ϕ and study its displacement convexity in Section 5. Section 6 is devoted to the functional inequalities and Section 7 is concerned with the concentration of measures. The gradient flow of H ϕ is studied in Sections 8, 9 in the compact and noncompact cases, respectively. We extend most results to Finsler manifolds in Section. Finally in Appendix, we compare our concentration of measures derived from the generalized Talagrand inequality with the Herbst-type argument deriving the concentration from the u ϕ -entropy inequality, which is a generalization of the logarithmic Sobolev inequality different from ours. ). 2 Preliminaries 2. Weighted Riemannian manifolds Throughout the article except Section,, g) will be an n-dimensional complete connected Riemannian manifold without boundary. As we are interested in the role of the curvature, we will always assume n 2. Denote by d g and vol g the Riemannian distance function and the Riemannian volume measure of, g). We fix an arbitrary measure ω = e f vol g, f C ), as our base measure. To control the behavior of ω, we modify the Ricci curvature Ric g of, g) as follows. Definition 2. Weighted Ricci curvature) Given N, ) [n, ], we define the N-Ricci curvature tensor of, ω) by Ric g + Hess g f if N =, Df Df Ric N := Ric g + Hess g f if N, ) n, ), N n Ric g + Hess g f Df Df) if N = n, where by convention =. We set Ric N v) := Ric N v, v) and will say that Ric N K holds for some K R if Ric N v) K v, v for every v T. Remark 2.2 The tensor Ric N was usually considered only for N [n, ], and then the monotonicity Ric N v) Ric N v) for N < N clearly holds. Note that Ric is the 6

7 famous Bakry Émery tensor and Ric N for N n, ) was introduced by Qian see [BE], [Qi] and [Lo] as well). Extending the range of N to, ) [n, ] violates the above monotonicity in N, however, observe that Ric N is non-decreasing in the parameter m := [ N ), n, where m := if N =. This observation will be helpful for understanding the validity of Theorem 5.7 below. Note that, if, ω) satisfies Ric N K for some K R and N [n, ), then it behaves like a Riemannian manifold with dimension bounded above by N and Ricci curvature bounded below by K see [Qi], [Lo], as well as [St2], [St3], [LV], [LV2], [Vi2, Part III] related to the curvature-dimension condition). For example, the following area growth inequality of Bishop type numerically extended to non-integer N s) holds. Denote by area ω [Sx, r)] the area of the sphere Sx, r) := {x d g x, x) = r} with respect to ω. Theorem 2.3 [Qi], [St3, Theorem 2.3]) If, ω) satisfies Ric N [n, ), then ) N R area ω [Sx, R)] area ω [Sx, r)] r holds for any < r < R and x. For N =, we have the following global estimate. for some N Theorem 2.4 [Vi2, Theorem 8.2]) Under the nonnegativity of Ric of, ω), exp λd g x, x) 2) dωx) < holds for any λ > and x. Though Theorems 2.3, 2.4 are generalized to Ric N K for K, we will need only the above special cases. 2.2 Wasserstein geometry Let us recall some basic notions and facts in optimal transport theory and Wasserstein geometry. See [AGS], [Vi] and [Vi2] for details and more information. Let X, d) be a metric space. A rectifiable curve γ : [, ] X is called a geodesic if it is locally minimizing and has a constant speed. We say that γ is minimal if it is globally minimizing, namely dγs), γt)) = s t dγ), γ)) holds for all s, t [, ]. A subset Y of X is said to be totally convex if, for any x, y Y, any minimal geodesic in X from x to y is contained in Y. For a complete Riemannian manifold, g), let P) be the set of all Borel probability measures on. Given µ P) and a measurable map T :, the push forward measure T µ of µ through T is defined by T µ[b] := µ[t B)] for all Borel sets B. For each p [, ), denote by P p ) P) the subset consisting of 7

8 measures µ of finite p-th moments, that is, d gx, x) p dµx) < for some and hence all) x. For µ, ν P), a probability measure π P ) is called a coupling of µ and ν if its projections are µ and ν, namely π[b ] = µ[b] and π[ B] = ν[b] hold for any Borel set B. We define the L p -Wasserstein distance between µ, ν P p ) by { } /p W p µ, ν) := inf d g x, y) p dπx, y)) π: a coupling of µ and ν. A coupling π is said to be optimal if it attains the infimum above. The function W p is indeed a distance function on P p ). The metric space P p ), W p ) is complete, separable and called the L p -Wasserstein space over. The Wasserstein space inherits several properties of. For instance, if is compact, then P p ), W p ) is also compact and the topology induced from W p coincides with the weak topology. We will mainly consider the quadratic case p = 2, and then we omit L 2 - and simply call W 2 and P 2 ), W 2 ) the Wasserstein distance function and the Wasserstein space. In view of optimal transport theory, W 2 µ, µ ) 2 is regarded as the least cost of transporting µ to µ, where the cost of transporting a unit mass from x to y is d g x, y) 2. A minimal geodesic µ t ) t [,] with respect to W 2 is then also called the optimal transport from µ to µ, and it can be described by using a family of minimal geodesics in the underlying space. We denote by Γ) the set of all minimal geodesics γ : [, ] endowed with the uniform topology induced from the distance function d Γ) γ, η) := sup t [,] d g γt), ηt)). For t [, ], the evaluation map ev t : Γ) is defined by ev t γ) := γt), which is clearly -Lipschitz. Proposition 2.5 [LV2, Proposition 2.], [Vi2, Corollary 7.22]) Given any minimal geodesic µ t ) t [,] P 2 ), there exists Π PΓ)) such that ev t ) Π = µ t for all t [, ] and that ev ev ) Π is an optimal coupling of µ and µ. In particular, for any totally convex set X of, d g ), P 2 X) is also totally convex in P 2 ), W 2 ). If one of µ and µ is absolutely continuous with respect to vol g, then a more precise description of a minimal geodesic µ t ) t [,] is obtained via the gradient vector field of a locally semi-convex function φ i.e., every point x admits a neighborhood on which φ is K-convex in the weak sense for some K R, see Definition 4.). For a measure ν on, we denote by P ac, ν) P) the subset of absolutely continuous measures with respect to ν. We also set P 2 ac, ν) := P 2 ) P ac, ν). Theorem 2.6 [FG, Theorem ]) Given any µ P 2 ac, vol g ) and µ P 2 ), there exists a locally semi-convex function φ : Ω R on an open set Ω with µ [Ω] = such that the map T t x) := exp x t φx)), t [, ], provides a unique minimal geodesic from µ to µ. Precisely, T T ) µ is a unique optimal coupling of µ and µ, and µ t := T t ) µ is a unique minimal geodesic from µ to µ with respect to W 2. If is compact, then the above theorem is due to ccann s celebrated work [c2] and we can take as the potential function φ a c-concave function for the cost cx, y) = 8

9 d g x, y) 2 /2. We do not give the definition of the c-concave function, what we need is only the fact that c-concave functions are locally semi-convex.) A locally semi-convex function is locally Lipschitz and twice differentiable almost everywhere by the Alexandrov Bangert theorem. Thus T t is differentiable µ -a.e. and the following Jacobian or onge Amperè) equation holds. Theorem 2.7 [Vi2, Theorems 8.7,.]) Under the same assumptions as Theorem 2.6 above, we have µ t P 2 ac, vol g ) for all t [, ). oreover, by putting ρ t ω := µ t = T t ) µ, J ω t x) := e fx) fttx)) det DT t x) ), we have ρ t T t x))j ω t x) = ρ x) and J ω t x) > for all t [, ) at µ -a.e. x Ω. In the case of ν P 2 ac, vol g ), the above assertions hold also at t =. Note that J ω t should be understood as the Jacobian with respect to ω, and its behavior is naturally controlled by the weighted Ricci curvature. This is a fundamental geometric intuition behind the curvature-dimension condition see Section 5). 2.3 Information geometry We briefly summarize some notions in information geometry associated with a nondecreasing, positive, continuous function ϕ :, ), ). We refer to [Na] and [Na2] for further discussion. We define the ϕ-logarithmic function on, ) by ln ϕ t) := t ϕs) ds, which is clearly strictly increasing. We will denote by l ϕ and L ϕ the infimum and the supremum of ln ϕ, that is, l ϕ := inf t> ln ϕt) = lim t ln ϕ t) [, ), L ϕ := sup ln ϕ t) = lim ln ϕ t), ]. t> t The inverse function of ln ϕ is called the ϕ-exponential function. function on R as if τ l ϕ, exp ϕ τ) := ln ϕ τ) if τ l ϕ, L ϕ ), if τ L ϕ. We also introduce the strictly convex function u ϕ r) := r ln ϕ t) dt, r [, ), provided that it is well-defined i.e., ln ϕ is integrable on, )). We extend it to the Lemma 2.8 The function u ϕ is well-defined if { } inf δ R s +δ is bounded on, ) <. 2.) ϕs) 9

10 Proof. As ϕ is positive and non-decreasing, it suffices to see u ϕ ) >. We deduce from the hypothesis that s/ϕs) is integrable on, ). This shows the claim since u ϕ ) = t ϕs) dsdt = s ds >. ϕs) Entropy is a function measuring the uncertainty of an event, and the divergence in information theory is a quantity expressing the difference between a pair of probability measures. In this spirit, the ϕ-entropy for ρω P ac, ω) is defined by E ϕ ρω) := u ϕ ρ) dω provided that it is well-defined). Then we define the Bregman divergence between ρω, σω P ac, ω) by { D ϕ ρω σω) := uϕ ρ) u ϕ σ) u ϕσ)ρ σ) } dω. 2.2) The strict convexity of u ϕ guarantees D ϕ ρω σω) > unless ρ = σ ω-a.e.. Furthermore, the square root of the divergence D ϕ satisfies a generalized Pythagorean theorem see [OW, Proposition 3]) and hence it can be regarded as a kind of distance function, though it is not symmetric i.e., D ϕ ρω σω) D ϕ σω ρω) in general). We define three more quantities measuring the order of ϕ for later use: { } s ϕs + t) ϕs) θ ϕ := sup lim sup ϕs) t t s > [, ], 2.3) { } s ϕs + t) ϕs) δ ϕ := inf lim sup ϕs) t t s > [, ), 2.4) { θ ϕ ) if θ ϕ, N ϕ := 2.5) if θ ϕ =. The following lemma will be useful. Lemma 2.9 The function s δ ϕ /ϕs) is non-increasing in s, ). oreover, if θ ϕ is finite, then the function s θϕ /ϕs) is non-decreasing in s, ). Proof. Assume θ ϕ < which will not play any role in the discussion on δ ϕ ), and fix s > and small ε >. By the definitions of θ ϕ and δ ϕ, there exists r ε s) > such that Consider the functions δ ϕ s ϕs) sup ϕs + t) ϕs) t,r εs)] t θ ϕ + ε 2. h + τ) := θ ϕ + ε 2 + τ)θ ϕ+ε, h τ) := δ ϕ + τ)δ ϕ ε τ τ

11 for τ >. Since h + and h are continuous and satisfy lim h + τ) = θ ϕ + ε τ 2 θ ϕ + ε) <, lim h τ) = δ ϕ δ ϕ ε) >, τ there exists τ ε > such that we have h + τ) < and h τ) > for all τ, τ ε ). Given any t, min{r ε s), sτ ε }), we have { s θ ϕ+ε s + t)θϕ+ε = tsθ ϕ+ε s ϕs + t) ϕs) + ts ) θϕ+ε } ϕs) ϕs + t) ϕs + t) ϕs) t ts tsθ ϕ+ε ϕs + t) h +ts ) <. As s > was arbitrary, this shows that s θϕ+ε /ϕs) is strictly increasing in s >. We similarly obtain s δϕ ε ϕs) s + t)δϕ ε ϕs + t) tsδϕ ε ϕs + t) h ts ) >, so that s δϕ ε /ϕs) is strictly decreasing. Letting ε, we complete the proof. Remark 2. The function ϕ will be sometimes normalized so as to satisfy ϕ) =. This costs no generality as we easily see the following relations for any a > : ln aϕ t) = a ln ϕ t), exp aϕ τ) = exp ϕ aτ), u aϕ r) = a u ϕ r), l aϕ τ) = a l ϕ, L aϕ = a L ϕ, θ aϕ = θ ϕ, δ aϕ = δ ϕ, N aϕ = N ϕ. 2.4 Information geometry continued: The case of ϕ m s) = s 2 m In [OT], we considered the power function ϕ m s) := s 2 m for m, 2] and the corresponding m-logarithmic and m-exponential functions. We have actually considered m [n )/n, ) in [OT], but ϕ m is non-decreasing only when m 2.) We summarize several facts in this especially important case. For brevity, we set l m := ln ϕm, e m := exp ϕm, l m := l ϕm, L m := L ϕm, θ m := θ ϕm, N m := N ϕm. A) In the case of ϕ s) = s, l and e coincide with the usual logarithmic and exponential functions, respectively. Thus we find l = and L =. We can easily observe θ = and N = as well. For ρω, σω P ac, ω), we deduce from u ϕ r) = r ln r r that E ϕ ρω) = ρ ln ρ dω +, D ϕ ρω σω) = ρ ln ρ σ dω. Namely E ϕ is the Boltzmann entropy up to adding, and D ϕ is the Kullback Leibler divergence. By choosing σ formally in the definition of D ϕ ρω σω), the relative entropy of µ P 2 ) with respect to ω is defined by lim ρ ln ρ dω if µ = ρω P Ent ω µ) := ε ac, {ρ ε} 2 ω), 2.6) otherwise.

12 In other words, the relative entropy is ) the Boltzmann entropy. B) For ϕ m s) = s 2 m with m, ), 2], l m and e m are given by power functions as l m t) = tm m, e mτ) = [ + m )τ] /m ) +, where we set [t] + := max{t, } and by convention a := for a <. Observe if m <, l m = if m <, L m = m if m >, m if m >, 2.7) θ m = 2 m and N m = m). As u ϕm r) = r m mr)/{mm )}, the ϕ m -entropy for ρω P ac, ω) is given by ρ m E ϕm ρω) = mm ) dω + m. Up to additive and multiplicative constants, this coincides with the Rényi-Tsallis) entropy S N ρω) := ρ N )/N dω 2.8) with N = N m, which is applied to complex strongly correlated) systems. The Bregman divergence between ρω, σω P ac, ω) is given by [ D ϕm ρω σω) = ρ m σ m ) mσ m ρ σ) ] dω. mm ) This coincides with the β-divergence, whose strength is its robustness. For instance, we refer to [TKE] for the roles and the differences of statistical divergences including the Bregman divergences. Note that as m we have l m t) l t), e m τ) e t), E ϕm ρω) E ϕ ρω), D ϕm ρω σω) D ϕ ρω σω). The function ϕ m = s 2 m is an extremal element among those ϕ s satisfying θ ϕ = 2 m in several respects, as one can see in the next useful lemma for instance. Lemma 2. Assume θ ϕ < 2 and put m = 2 θ ϕ. Then for any t > and r R we have Proof. It follows from Lemma 2.9 that, for any t >, ϕ) t ϕ) l mt) ln ϕ t) tθϕ ϕt) l mt), 2.9) s θϕ ds exp ϕ r) e m ϕ)r ). 2.) t tθϕ ds ϕs) ϕt) 2 t s θϕ ds.

13 This is exactly 2.9) since θ ϕ = 2 m. As for 2.), the assertion for r l ϕ is trivial since exp ϕ r) = by definition. If r L ϕ, then we deduce from 2.9) that ϕ)r ϕ)l ϕ L m, which shows e m ϕ)r) =. We therefore assume l ϕ < r < L ϕ and set t := exp ϕ r) >. Then we obtain again from 2.9) that exp ϕ r) = t = e m lm t) ) e m ϕ) lnϕ t) ) = e m ϕ)r ). Taking the limits as t or t in 2.9), we obtain from 2.7) the following. Lemma 2.2 Suppose θ ϕ < 2. i) If θ ϕ <, then l ϕ > equivalently, if l ϕ =, then θ ϕ ). ii) If θ ϕ, then L ϕ = equivalently, if L ϕ <, then θ ϕ > ). We similarly find the corresponding estimates concerning δ ϕ. Note that δ ϕm = θ m = 2 m. Lemma 2.3 Assume δ ϕ < 2 and put m = 2 δ ϕ. Then for any t > and r R we have t δϕ ϕt) l mt) ln ϕ t) ϕ) l ) mt), exp ϕ r) e m ϕ)r. In particular, i) If δ ϕ >, then L ϕ < equivalently, if L ϕ =, then δ ϕ ). ii) If δ ϕ, then l ϕ = equivalently, if l ϕ >, then δ ϕ < ). 3 Displacement convexity classes DC N In this section, we introduce the important classes of convex functions. These classes were first considered by ccann [c] for N see also [LV2, 5.], [Vi, Section 5.2], [Vi2, Chapter 6]), we adopt the same definition also for N <. Definition 3. Displacement convexity classes) For N, ) [, ), we define DC N as the set of all continuous convex functions u : [, ) R such that u) = and that the function ψ N r) := r N ur N ) is convex on, ). In a similar way, DC is defined as the set of all continuous convex functions u : [, ) R such that u) = and that the function is convex on R. ψ r) := e r ue r ) The following is well-known for N, we give a proof for completeness. 3

14 Lemma 3.2 If u DC N for N [, ] resp. N, )), then the function ψ N is non-increasing resp. non-decreasing). Proof. For N [, ) and < s < t, the convexity of u and u) = yield { } ψ N t) = t N ut N ) t N )u) sn + sn t N t N us N ) = ψ N s). We similarly obtain for N = and s, t R with s < t that { } ψ t) = e t ue t ) e t )u) es + es e t e t ue s ) = ψ s). Finally, for N < and < s < t, it holds { } ψ N s) = s N us N ) s N )u) s N + s N t N t N ut N ) = ψ N t). It is also known that DC N DC N for N < N. This monotonicity in N is violated by extending to N <, but the monotonicity in m = N )/N [, ) holds instead. Compare this with the monotonicity of Ric N in m Remark 2.2). Lemma 3.3 For each N, N, ) [, ] with m < m, we have DC N DC N, where we set m = N )/N, m = N )/N and m = if N = resp. m = if N = ). Proof. We first consider the case of m < m < equivalently, N < N < ). For any u DC N and r >, we observe ψ N r) = r N ur N ) = r N/N ) N u r N/N ) N ) = ψn r N/N ). This is convex in r since the function r r N/N is concave and ψ N is convex and nonincreasing. Thus u DC N and hence DC N DC N. The other cases are similar. For < m < m, we have N < N < so that r r N/N is convex and ψ N is non-decreasing. When m = > m, ψ N r) = ψ N log r) holds and note that r N log r is concave and ψ is non-increasing. For m = < m, we find ψ r) = ψ N e r/n ) and that r e r/n is convex and ψ N is non-decreasing. We shall write down a condition for u ϕ DC N on ϕ. As u ϕ is continuous, convex and satisfies u ϕ ) = by definition once it is well-defined, it is sufficient to check 2.) and the convexity of ψ N. Proposition 3.4 Assume that ϕ satisfies the condition 2.). Then the function ψ N for N, ), ] is convex if and only if t s ϕs) ds N t 2 N ϕt) holds for all t >, where N/N ) = if N =. 3.) 4

15 Proof. We first of all recall that u ϕ is well-defined thanks to 2.) Lemma 2.8). For N, ), ) and r >, we calculate ψ Nr) = Nr N u ϕ r N ) Nr u ϕr N ), ψ Nr) = NN )r N 2 u ϕ r N ) + N N 2 )r 2 u ϕr N ) + N 2 r N 2 u ϕr N ) { = NN )r N 2 u ϕ r N ) r N ln ϕ r N ) + N } r 2N. N ϕr N ) Note that NN ) >. For any t >, we have t u ϕ t) t ln ϕ t) + N t 2 N ϕt) = {ln ϕ τ) ln ϕ t)} dτ + N t 2 N ϕt) t t = ϕs) dsdτ + N t 2 t N ϕt) = s ϕs) ds + N t 2 N ϕt). τ Therefore ψ N if and only if 3.) holds. For N =, we similarly obtain { } ψ r) = e r u ϕ e r ) e r ln ϕ e r ) + e 2r ϕe r ) for r R, and for t >. u ϕ t) t ln ϕ t) + t t2 ϕt) = s ϕs) ds + t2 ϕt) Theorem 3.5 If θ ϕ < 2, then the condition 2.) holds and we have u ϕ DC Nϕ. Proof. We deduce from Lemma 2.9 that sθ ϕ ϕs) ϕ) for all s, ), this implies 2.) since θ ϕ < 2. Lemma 2.9 also yields that, for any t >, t s t ϕs) ds t θ ϕ ϕt) s θ ϕ ds = t 2 2 θ ϕ ϕt). This is nothing but 3.) with N = N ϕ recall the definition of N ϕ in 2.5)), and hence u ϕ DC Nϕ by Proposition 3.4. Recall that ϕ m s) = s 2 m with m, 2] satisfies θ m = 2 m < 2. Hence Theorem 3.5 shows u ϕm DC Nm. We close the section with a partial converse of Theorem 3.5. Proposition 3.6 If the condition 2.) holds, δ ϕ < 2 and if we have u ϕ DC N with some N, ), ], then it holds δ ϕ N + )/N where N + )/N = for N = ). 5

16 Proof. Lemma 2.9 with Proposition 3.4 yields that, for any t >, N t 2 N ϕt) t s t ϕs) ds t δ ϕ ϕt) s δ ϕ ds = t 2 2 δ ϕ ϕt), which shows δ ϕ N + )/N as desired. In particular, for m, 2], u ϕm DC N if and only if m N )/N. 4 Admissible spaces This section is devoted to introducing the class of spaces admissible in our consideration. Recall our weighted Riemannian manifold, ω) and a function ϕ as in Subsection 2.3. From here on, we further fix the reference measure where Ψ C) such that ν = σω := exp ϕ Ψ)ω, Ψ > L ϕ on, Ψ ϕ := Ψ L ϕ, l ϕ ) ). 4.) Note that supp ν = ϕ Ψ. For later convenience, let us define the K-convexity of a function on a general metric space. Definition 4. K-convexity) Given K R, we say that a function Ψ : X, ] on a metric space X, d) is K-convex in the weak sense denoted by Hess Ψ K by slight abuse of notation) if it is not identically and, for any two points x, y X, there exists a minimal geodesic γ : [, ] X from x to y along which holds for all t [, ]. Ψ γt) ) t)ψx) + tψy) K 2 t)tdx, y)2 4.2) We remark that, on a Riemannian manifold, 4.2) certainly holds for any minimal geodesic γ : [, ] by approximation. Indeed, γ [ε, ε] is a unique minimal geodesic for any ε > and Ψ is continuous. We are interested in the situation that Ric Nϕ as well as Hess Ψ K hold see Theorem 5.7). Finer analysis is possible in the particular case of K > Sections 6, 7). We prove a lemma in such a case for later use. The open ball of center x and radius r > will be denoted by Bx, r). Lemma 4.2 Suppose that ϕ) = this costs no generality, see Remark 2.), Hess Ψ K for some K >, and take a minimizer x of Ψ. i) If l ϕ >, then the set Ψ ϕ as in 4.) is totally convex and Ψ ϕ Bx, R) holds with R = 2l ϕ + Ψx ))/K. oreover, supp ν is also totally convex and compact. 6

17 ii) If l ϕ =, N ϕ [n, ) and if Ric Nϕ, then we have ϕ Ψ =, ν[] < and d g x, x) p σx) a dωx) C ν[] a + C 2 K an ϕ < for any a /2, ] and p [, ) satisfying 2a )N ϕ p >, where C = C ω) and C 2 = C 2 a, p, θ ϕ, ω). In particular, σ L a, ω) for all a /2, ] and ν[] ν P p ac, ω) for all p [, N ϕ ). iii) If N ϕ = and Ric Nϕ, then σ L a, ω) for any a >. Proof. We first remark that the assumption Hess Ψ K > guarantees the unique existence of a point x ϕ Ψ such that Ψx ) = inf Ψ. We deduce from the K-convexity 4.2) that Ψ γ) ) Ψx ) + K 2 d g x, γ) ) 2 4.3) holds for all minimal geodesics γ : [, ] with γ) = x. i) For any minimal geodesic γ : [, ] connecting two points x, y Ψ ϕ, we have Ψ γt) ) t)ψx) + tψy) K 2 t)td gx, y) 2 < l ϕ, so that γ is contained in ϕ Ψ. The total convexity of supp ν can be seen similarly. Precisely, for any x, y Ψ L ϕ, l ϕ ]), γ as above satisfies γ, )) ϕ Ψ. This in fact implies that supp ν = ϕ Ψ = Ψ L ϕ, l ϕ ]) and is totally convex. We moreover obtain ϕ Ψ Bx, R) from 4.3), and thus supp ν is compact. ii) The first assertion ϕ Ψ = is obvious by definition see 4.)). Note that N ϕ [n, ) implies θ ϕ, n + )/n] see 2.5)). Set m := 2 θ ϕ < and take a /2, ] and p satisfying 2a )N ϕ p >. Then 2.) and 4.3) imply d g x, x) p σx) a dωx) σ a dω + e m Ψx ) K ) a 2 r2 r p area ω [Sx, r)] dr. Bx,) We mention that m = N ϕ )/N ϕ and, for s < L ϕ and t <, we have e m s + t) = {e m s) m + m )t} N ϕ. Thanks to the hypothesis Ric Nϕ with N ϕ [n, ), we can estimate the second term 7

18 by Theorem 2.3 as e m Ψx ) K 2 r2 ) a r p area ω [Sx, r)] dr area ω [Sx, )] = area ω [Sx, )] { e m Ψx ) ) m + m) K 2 r2 } anϕ r p+n ϕ dr { e m Ψx ) ) m r 2 + m) K 2 } anϕ r 2a )N ϕ+p dr { area ω [Sx, )] m) K } anϕ r 2a )Nϕ+p dr 2 { = 2a )N ϕ p area ω[sx, )] m) K } anϕ. 2 We used the condition 2a )N ϕ p > in the last equality. Choosing a = and p =, we in particular find ν[] <. Then the Hölder inequality yields that a σ a dω σ dω) ω[bx, )] a ν[] a ω[bx, )] a. Bx,) Bx,) Thus choosing C := max{ω[bx, )], } ω[bx, )] a, ) anϕ 2 C 2 := area ω [Sx, )] 2a )N ϕ p m gives the desired estimate. iii) Combining 4.3) with 2.) provides, as θ ϕ =, σx) a dωx) exp ϕ Ψx ) K ) a 2 d gx, x) 2 dωx) exp aψx ) ) exp ak2 ) d gx, x) 2 dωx). Hence the assertion follows from Theorem 2.4. Now we introduce the conditions for a quadruple, ω, ϕ, Ψ) to be admissible in our consideration. Definition 4.3 Admissibility) We say that a quadruple, ω, ϕ, Ψ) is admissible if all the following conditions hold: A-) ϕ) =. A-2) N ϕ, ] [n, ] and N ϕ 2 or, equivalently, θ ϕ [, n + )/n] and θ ϕ < 3/2. A-3) Ψ > L 2 θϕ on and Ψ ϕ = Ψ L ϕ, l ϕ )). 8

19 A-4) h ϕ σ) L, ω) and σ ln ϕ σ) L, ω), where h ϕ := u ϕ if L ϕ = and h ϕ r) := u ϕ r) rl ϕ if L ϕ < see 5.3) below). We mention that l m l ϕ and L m L ϕ hold with m = 2 θ ϕ by 2.9), and hence Ψ ϕ Ψ ϕ m by A-3). The first condition ϕ) = is merely the normalization see Remark 2.), and A-4) is imposed for ν being adopted as a reference measure of the Bregman divergence see 2.2)). The next lemma ensures that A-4) automatically holds if Ric Nϕ and Hess Ψ K for some K >. Lemma 4.4 Suppose that, ω, ϕ, Ψ) satisfies A-), A-2), Ψ > L ϕ on, Ψ ϕ, Ric Nϕ and Hess Ψ K for some K >. Then A-4) also holds. Proof. The case of l ϕ > is clear due to Lemma 4.2i), so that we assume l ϕ = and then θ ϕ Proposition 2.2i)). Observe also that u ϕ σ) L, ω) implies h ϕ σ) L, ω) since ν[] < by Lemma 4.2ii), iii). Let x be the minimizer of Ψ and set R := max{, 2Ψx )}/K. Note that the K-convexity of Ψ 4.3) guarantees that, on \ Bx, R), σ = exp ϕ Ψ) exp ϕ Ψx ) K ) 2 R2 exp ϕ ) =. We first consider the case of θ ϕ >. On \ Bx, R), 2.9) implies that σ ln ϕ σ) = σ ln ϕ σ) σl 2 θϕ σ) = N ϕ σ 2 θ ϕ σ), σ σ u ϕ σ) = ln ϕ t) dt N ϕ t θ ϕ σ 2 θ ϕ ) ) dt = N ϕ σ. 2 θ ϕ Thus we have σ ln ϕ σ) dω u ϕ σ) dω Bx,R) Bx,R) σ ln ϕ σ) dω + u ϕ σ) dω + \Bx,R) \Bx,R) N ϕ σ 2 θ ϕ σ) dω, σ 2 θ ϕ ) N ϕ σ 2 θ ϕ As 2 θ ϕ /2, ) by θ ϕ < 3/2, Lemma 4.2ii) ensures u ϕ σ), σ ln ϕ σ) L, ω). In the case of θ ϕ =, we similarly have on \ Bx, R) dω. σ ln ϕ σ) σ ln σ σ, u ϕ σ) = Then the claim follows from Lemma 4.2iii). σ ln ϕ t) dt σ t dt = 2 σ. We close the section with an auxiliary lemma on how to normalize ν when K >. Lemma 4.5 Let, ω, ϕ, Ψ) be admissible, Ric Nϕ and Hess Ψ K for some K >, and set I = l, L) := l ϕ + inf Ψ, L ϕ + inf Ψ). We in addition assume that Ψ is differentiable at the minimizer of Ψ if N ϕ = n. Then there exists some λ I such that exp ϕ λ Ψ)ω P ac, ω). 9

20 Proof. We first remark that inf Ψ >, and that for any λ I ) Ψ λ > inf Ψ L ϕ + inf Ψ = L ϕ as well as HessΨ λ) = Hess Ψ K hold. Thus Ξλ) := exp ϕ λ Ψ) dω < by Lemma 4.2. Since Ξ is non-decreasing and continuous on I by Lebesgue s dominated convergence theorem or the monotone convergence theorem), we are done if lim λ l Ξλ) < < lim λ L Ξλ) holds. We also deduce from the dominated convergence theorem that lim λ l Ξλ) =. If L ϕ =, then we find lim λ Ξλ) = by the monotone convergence theorem. The rest is to prove lim λ L Ξλ) > when L ϕ <. Note that L ϕ < implies lim s ϕs) = by definition, and θ ϕ > i.e., N ϕ [n, )) by Proposition 2.2ii). Let x be the unique minimizer of Ψ and take R > such that Bx, R ) ϕ Ψ and that Bx, R ) contains no cut point of x. Then, for any x Sx, r) with < r < R R, the K-convexity of Ψ provides Ψx) where we set r R ) Ψx ) + r R a = ar) := R sup Sx,R) Ψ K 2 r R) r R R2 = Ψx ) + K 2 r2 + ar, 4.4) ) sup Ψ Ψx ) K Sx,R) 2 R2. Observe that lim R ar) < by the K-convexity of Ψ, and that lim R ar) = holds if N ϕ = n since Ψ is assumed to be differentiable at x. In both cases N ϕ > n or N ϕ = n) we can choose R, R ] small enough to satisfy Then take large λ I such that K 2 R2 + ar < L ϕ, 2a < ) /Nϕ areaω [Sx, R)]. 4.5) N ϕ R N ϕ λ > Ψx ) + K 2 R2 + ar. Set e λ ϕr) := exp ϕ λ Ψx ) K ) 2 r2 ar and note that it is decreasing. We deduce from Theorem 2.3 and 4.4) that Ξλ) exp ϕ λ Ψ) dω Bx,R) area ω[sx, R)] R Nϕ = area ω[sx, R)] R Nϕ R R N ϕ R r N ϕ e λ ϕr) dr N ϕ e λ ϕr) 2 area ω [Sx, r)]e λ ϕr) dr R ) r N ϕ e λ N ϕ) r) dr. ϕ

21 The concavity of ln ϕ yields and hence it holds for t := e λ ϕr) that Kr = a + ln ϕ t) ln ϕ e λ ϕ ) ) + t eλ ϕ) ϕe λ ϕ)), a 2 + 2K { ln ϕ e λ ϕ ) ) ln ϕ t) } a + a 2 + 2K eλ ϕ) t ϕe λ ϕ)). By the change of variables formula for t = e λ ϕr), we have R r N ϕ e λ ϕ) r) dr = K Nϕ K N ϕ Combining the triangle inequality e λ ϕ ) e λ ϕ R) e λ ϕ ) e λ ϕr) a + a 2 + 2K { ln ϕ e λ ϕ ) ) ln ϕ t) }) N ϕ dt a + a 2 + 2K eλ ϕ) t ϕe λ ϕ)) ) Nϕ dt. { e λ ϕ ) ) /Nϕ a + a 2 + 2K eλ ϕ) t Nϕ dt} e ϕe λ ϕr) λ ϕ)) { e λ ϕ ) ) } /Nϕ a 2 + 2K eλ ϕ) t Nϕ dt {e λ ϕe ϕ) e λ ϕr)} /N ϕ a λ ϕ)) e λ ϕr) with Jensen s inequality for the convex function s a 2 + s) Nϕ s ), we deduce that ) Nϕ e λ ϕ ) a + a 2 + 2K eλ ϕ) t dt e ϕe λ ϕr) λ ϕ)) { {e λ ϕ) e λ ϕr)} a 2 + 2K } Nϕ e λ ϕ ) e λ ϕ) t ϕe λ ϕ)) e e λ ϕr) λ ϕ) e λ ϕr) dt a = {e λ ϕ) e λ ϕr)} a 2 + K eλ ϕ) e λ ϕr) ϕe λ ϕ)) a) Nϕ. 2

22 Hence we obtain, as ϕs) s θ ϕ for s by Lemma 2.9, lim Ξλ) area ω[sx, R)] λ L N ϕ R Nϕ K lim {e λ ϕ) e λ ϕr)} Nϕ λ L area ω[sx, R)] N ϕ R N ϕ K N ϕ = area ω[sx, R)] N ϕ R N ϕ K N ϕ = area ω[sx, R)] N ϕ R N ϕ K N ϕ lim s s a 2 + a 2 + K eλ ϕ) e λ ϕr) ϕe λ ϕ)) ) Nϕ Ks s + e L ϕr)) a θ ϕ [s lim θϕ ] Nϕ a 2 + Ks θ ϕ a) s a2 + Kt a lim t t a ) Nϕ = area ω[sx, R)] N ϕ R N ϕ 2a) N ϕ. ) Nϕ This is bigger than by the choice of R recall 4.5)) and we complete the proof. Remark 4.6 If Ric Nϕ and Hess Ψ K for some K >, then Lemma 4.2 yields ν[] <. We will sometimes normalize, ω, ϕ, Ψ) so as to satisfy ν[] = Sections 6, 7). There are two ways of such a normalization: Put a := ν[] and consider, ω, ϕ, Ψ) :=, aω, ϕ, Ψ), i.e., ω = e f+ln a vol g. Take λ as in Lemma 4.5 and consider, ω, ϕ, Ψ) :=, ω, ϕ, Ψ λ). In both cases it is easily seen that the conditions Ric Nϕ and Hess Ψ K are preserved. These two normalizations are equivalent when ϕ = ϕ, where we indeed observe exp Ψ) ω = exp Ψ f + ln a)ω, exp Ψ)ω = exp Ψ f + λ)ω, and hence λ = ln a. 5 ϕ-relative entropy and its displacement convexity In this section, we introduce a generalization of the relative entropy, that we call the ϕ-relative entropy, associated with functions ϕ in an appropriate class. For the relative entropy on a unweighted) Riemannian manifold, it is known by von Renesse and Sturm [vrs] that its K-convexity in the Wasserstein space P 2 ), W 2 ) in the sense of Definition 4.) is equivalent to the lower Ricci curvature bound Ric K. Then it was shown by Lott and Villani [LV2] that Ric K further implies a kind of convexity property of a class of entropies including the relative entropy. In this sense, the relative entropy is an extremal element in such a class of entropies. In the same spirit, our main theorem in the section Theorem 5.7) asserts that the m-relative entropy induced from ϕ = ϕ m studied in [OT], recall Subsection 2.4 as well) is an extremal one in an appropriate class of ϕ-relative entropies. 22

23 5. Curvature-dimension condition We begin with a brief review of Lott, Sturm and Villani s curvature-dimension condition. To formulate it, we need to introduce the two functions often used in comparison theorems in Riemannian geometry. For K R, N, ) and < r < π N )/K if K > ), we consider the function s K,N r) := N )/K sinr K/N )) if K >, r if K =, N )/K sinhr K/N )) if K <. This is the solution to the differential equation s K,N + K N s K,N = with the initial conditions s K,N ) = and s K,N ) =. For n N with n 2, s K,nr) n is proportional to the area of the sphere of radius r in the n-dimensional space form of constant sectional curvature K/n ). Using s K,N, we define β t K,Nr) := ) N sk,n tr), β t ts K,N r) K, r) := e K t2 )r 2 /6 for K, N, r as above and t, ). Now, we are ready to state Sturm s curvaturedimension condition characterizing lower Ricci curvature bounds, developed in [vrs], [St] and [St3] in gradually general situations see also [CS], [CS2] for related important work). Recall 2.8) and 2.6) for the definitions of the Rényi entropy S N and the relative entropy Ent ω. Theorem 5. Sturm s curvature-dimension condition) Let, ω) be a weighted Riemannian manifold. We have Ric N K for some K R and N [n, ) if and only if any pair of measures µ = ρ ω, µ = ρ ω Pac, 2 ω) satisfies S N µ t ) t) βk,n t dg x, y) ) /N ρ x) /N dπx, y) t βk,n t dg x, y) ) /N ρ y) /N dπx, y) 5.) for all t, ), where µ t ) t [,] P 2 ac, ω) is the unique minimal geodesic from µ to µ, and π is the unique optimal coupling of µ and µ. Similarly, Ric K is equivalent to the K-convexity of Ent ω, Ent ω µ t ) t) Ent ω µ ) + t Ent ω µ ) K 2 t)tw 2µ, µ ) 2. For K =, we find β t,n and 5.) is nothing but the convexity of S N, S N µ t ) t)s N µ ) + ts N µ ). 23

24 For K, however, 5.) is not simply the K-convexity of S N. Lott and Villani s version of the curvature-dimension condition requires a similar convexity condition, but for all entropies induced from functions in DC N [LV], [LV2], [Vi2, Part III]). For U DC N and µ P 2 ), we denote by µ = ρω + µ s its Lebesgue decomposition into absolutely continuous and singular parts with respect to the base measure ω, and define U ω µ) := lim Uρ) dω + U ε {ρ ε} )µ s [], U Ur) ) := lim. r r We set = by convention, and remark that U ) = lim r U r) holds due to the convexity of U. Theorem 5.2 Lott and Villani s version) We have Ric N K for some K R and N [n, ] if and only if, given any pair of measures µ, µ P 2 ) decomposed as µ i = ρ i ω + µ s i i =, ), there is a minimal geodesic µ t ) t [,] P 2 ) between them such that U ω µ t ) t) + t βk,n t dg x, y) ) U β t K,N dg x, y) ) U ρ x) β t K,N d gx, y)) ρ y) β t K,N d gx, y)) ) dπ x y)dωx) ) dπ y x)dωy) + U ){ t)µ s [] + tµ s []} 5.2) holds for all U DC N and t, ), where π x P) µ -a.e. x) and π y P) µ -a.e. y) denote the disintegrations of π by µ and µ, i.e., dπx, y) = dπ x y)dµ x) = dπ y x)dµ y). Recall that DC N DC N for n N < N Lemma 3.3). This agrees with the monotonicity Ric N Ric N for n N N. In the case where both µ and µ are absolutely continuous with respect to ω, we find dπx, y) = ρ x)dπ x y)dωx) = ρ y)dπ y x)dωy) and 5.2) is rewritten in the more symmetric form U ω µ t ) t) + t β t K,N d gx, y)) ρ x) U β t K,N d gx, y)) ρ y) U ρ x) β t K,N d gx, y)) ρ y) β t K,N d gx, y)) ) dπx, y) ) dπx, y). It is in fact enough to consider only absolutely continuous measures, the general inequality 5.2) follows by approximating µ, µ P 2 ) with absolutely continuous ones. In this sense the essential difference between Theorems 5. and 5.2 is their choices of admissible entropies. 24

25 Besides Riemannian manifolds, these two versions of the curvature-dimension condition are equivalent for metric measure spaces where geodesics do not branch, such as Finsler manifolds and Alexandrov spaces or for spaces with Riemannian Ricci curvature bounded below in the sense of [AGS3] which are essentially non-branching, see recent [RS]). In other words, Sturm s version implies Lott and Villani s one. Roughly speaking, this implication can be seen by localizing Sturm s 5.) thanks to the non-branching property, and then integrating these local inequalities for each U DC N yields 5.2). The same infinitesimal estimate Claim 5.8) will appear in our discussion. Theorem 5.2 is extended to general Finsler manifolds by introducing the appropriate notion of the weighted Ricci curvature see Section and [Oh2]). General not necessarily differentiable) metric measure spaces satisfying the condition in Theorem 5. or 5.2 are known to behave like Riemannian manifolds of Ric K and dim N in geometric and analytic respects [St2], [St3], [LV], [LV2], [Vi2, Part III]). We shall generalize this technique to ϕ-relative entropies in the following sections but only on Riemannian or Finsler manifolds). 5.2 ϕ-relative entropy H ϕ Let, ω, ϕ, Ψ) be an admissible space in the sense of Definition 4.3. We modify u ϕ as, for r, { u ϕ r) if L ϕ =, h ϕ r) := 5.3) u ϕ r) rl ϕ if L ϕ <. We also define h ϕ ) := lim r h ϕr) = { if L ϕ =, if L ϕ <. Note that u ϕ DC Nϕ by Theorem 3.5 thanks to the admissibility) immediately implies h ϕ DC Nϕ. oreover, if L ϕ <, then h ϕ is non-increasing and hence nonpositive. We set L ϕ, ω) := {ρ : R measurable, h ϕ ρ) L, ω)}, P Ψ ) := {µ P) Ψ L Ψ ϕ, µ)} we will use these notations only in Remark 5.4). Now the Bregman divergence 2.2) leads us to the following generalization of the relative entropy. Definition 5.3 ϕ-relative entropy) Given µ P), letting µ = ρω + µ s be its Lebesgue decomposition, we define the ϕ-relative entropy of µ by H ϕ µ) := {h ϕ ρ) h ϕσ)ρ} dω h ϕσ) dµ s + h ϕ )µ s [] = h ϕ ρ) dω h ϕσ) dµ + h ϕ )µ s [] 5.4) if h ϕ ρ) L, ω) and h ϕσ) L, µ), otherwise we set H ϕ µ) :=. 25

26 Let us summarize several remarks on Definition 5.3. Remark 5.4 ) In the second term of 5.4), to be precise, we set { h l ϕ if L ϕ =, ϕσ) := on \ ϕ Ψ. l ϕ L ϕ if L ϕ < This causes no problem because = ϕ Ψ if l ϕ =. The additional condition µ[ϕ Ψ ] = in other words, µ Pϕ Ψ )) will be imposed only when we compare the behavior of Ψ with that of H ϕ as in Theorems 5.7, 8.7 and so forth). 2) We remark that the condition A-4) in the admissibility guarantees that σ L ϕ, ω) as well as Ψ L ϕ Ψ, ν). Thus we have H ϕ ν) R by extending the definition 5.4) verbatim). 3) The validity of the definition of h ϕ ) for the lower semi-continuity of H ϕ, see Lemma 5.6) would be understood by the following observation: For small ε >, put µ ε = ρ ε ω := ω[bx, ε)] χ Bx,ε) ω, where χ Bx,ε) stands for the characteristic function of Bx, ε). Then we have ) h ϕ ρ ε ) dω = ω[bx, ε)] h ϕ h ω[bx, ε)] ϕ ) Bx,ε) as ε tends to zero. 4) Finally, as for the domain of H ϕ, it is more consistent to set H ϕ µ) = only if h ϕ ρ) ρh ϕσ) L, ω). However, as we will sometimes treat the internal energy h ϕρ) dω and the potential energy h ϕσ) dµ separately, we consider the smaller domain in Definition 5.3. This may cause a problem when considering the lower semicontinuity of H ϕ, whereas we need it only for compact see Lemma 5.6 below) where h ϕσ) L, µ) is always true so that h ϕ ρ) L, ω) if and only if h ϕ ρ) ρh ϕσ) L, ω)). Let us add a comment on the relation between ρ L ϕ, ω) and µ P Ψ ). Assume L ϕ < and µ = ρω + µ s P Ψ ). The nonpositivity and the convexity of h ϕ yield h ϕ ρ) dω = h ϕ ρ) dω {h ϕ σ) + h ϕσ)ρ σ)} dω = {h ϕ σ) h ϕσ)σ} dω ρh ϕσ) dω <. Hence ρ L ϕ, ω) automatically holds. One can also see the converse implication ρ L ϕ, ω) µ P Ψ )) for the special case ϕ m s) = s m with m >, where L m = as in 2.7) [OT, Remark 3.22)]). It is easily observed that ν is a unique ground state of H ϕ provided that ν[] = ). Lemma 5.5 Suppose ν[] =. For any µ = ρω + µ s P), we have H ϕ µ) H ϕ ν) and equality holds if and only if µ = ν. 26

27 Proof. We assume H ϕ µ) < without loss of generality. Observe that H ϕ µ) H ϕ ν) = {h ϕ ρ) h ϕ σ) h ϕσ)ρ σ)} dω On the one hand, if µ s [] >, then the singular part h ϕσ) dµ s + h ϕ )µ s [] h ϕσ) dµ s + h ϕ )µ s []. 5.5) in 5.5) is positive if L ϕ < recall that h ϕσ) = l ϕ L ϕ < on \ ϕ Ψ ) or infinity if L ϕ =. On the other hand, the strict convexity of h ϕ implies that the absolutely continuous part {h ϕ ρ) h ϕ σ) h ϕσ)ρ σ)} dω in 5.5) is nonnegative and equality holds if and only if ρ = σ ω-a.e.. Thus H ϕ µ) H ϕ ν) and equality holds if and only if µ s [] = and ρ = σ holds ω-a.e.. Observe from 5.5) that it holds D ϕ µ ν) = H ϕ µ) H ϕ ν) for any absolutely continuous measure µ with respect to ω. Thus the Bregman divergence D ϕ µ ν) measures the difference between the entropies at µ and the ground state ν. In [OT], we have studied the specific function ϕ m s) = s 2 m and the associated m-relative entropy H m µ ν) for m [n )/n, ), ). In the present context, H m µ ν) coincides with H ϕm µ) H ϕm ν). The following lemma will be used in Section 8 Claim 8.8) to construct a discrete approximation of the gradient flow of H ϕ, where is assumed to be compact. Lemma 5.6 Let be compact. Then the ϕ-relative entropy H ϕ is lower semi-continuous with respect to the weak topology, that is to say, if a sequence {µ i } i N P) weakly converges to µ P), then we have H ϕ µ) lim inf i H ϕµ i ). Proof. We divide H ϕ µ) into two parts as H ϕ ) µ) := h ϕ ρ) dω + h ϕ )µ s [], H ϕ 2) µ) := h ϕσ) dµ, where µ = ρω + µ s. Note that h ϕσ) < thanks to the compactness of recall Remark 5.4)). Then H ϕ 2) µ) is clearly continuous in µ and the lower semi-continuity of H ϕ ) µ) follows from [LV2, Theorem B.33]. 5.3 Displacement convexity of H ϕ In our previous work [OT], we showed that the displacement K-convexity of the m-relative entropy H m µ ν) = H ϕm µ) H ϕm ν) with respect to µ P 2 ) is equivalent to the 27