RICCI CURVATURE OF FINITE MARKOV CHAINS VIA CONVEXITY OF THE ENTROPY

Transcription

1 RICCI CURVATURE OF FINITE MARKOV CHAINS VIA CONVEXITY OF THE ENTROPY MATTHIAS ERBAR AND JAN MAAS Abstract. We study a new notion of Ricci curvature that applies to Markov chains on discrete spaces. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott, Sturm, and Villani for geodesic measure spaces. In order to apply to the discrete setting, the role of the Wasserstein metric is taken over by a different metric, having the property that continuous time Markov chains are gradient flows of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry Émery and Otto Villani. Furthermore we show that Ricci curvature bounds are preserved under tensorisation. As a special case we obtain the sharp Ricci curvature lower bound for the discrete hypercube. Contents. Introduction. The metric W 7 3. Geodesics 7 4. Ricci curvature 9 5. Examples 5 6. Basic Constructions 9 7. Functional Inequalities 3 References 37. Introduction In two independent contributions Sturm [4] and Lott and Villani [7] solved the long-standing open problem of defining a synthetic notion of Ricci curvature for a large class of metric measure spaces. The key observation, proved in [39], is that on a Riemannian manifold M, the Ricci curvature is bounded from below by some constant κ R, if and only if the Boltzmann Shannon entropy H(ρ) = ρ log ρ dvol is κ- convex along geodesics in the L -Wasserstein space of probability measures on M. The latter condition does not appeal to the geometric structure of M, but only requires a metric (to define the L -Wasserstein metric W ) and a reference measure (to define the entropy H). Therefore this condition can be used in order to define a notion of Ricci curvature lower boundedness on more Date: July 7,. JM is supported by Rubicon subsidy of the Netherlands Organisation for Scientific Research (NWO).

2 MATTHIAS ERBAR AND JAN MAAS general metric measure spaces. This notion turns out to be stable under Gromov Hausdorff convergence and it implies a large number of functional inequalities with sharp constants. The theory of metric measure spaces with Ricci curvature bounds in the sense of Lott, Sturm, and Villani is still under active development [, 3]. However, the condition of Lott Sturm Villani does not apply if the L - Wasserstein space over X does not contain geodesics. Unfortunately, this is the case if the underlying space is discrete (even if the underlying space consists of only two points). The aim of the present paper is to develop a variant of the theory of Lott Sturm Villani, which does apply to discrete spaces. In order to circumvent the nonexistence of Wasserstein geodesics, we replace the L -Wasserstein metric by a different metric W, which has been introduced in [8]. There, it has been shown that the heat flow associated with a Markov kernel on a finite set is the gradient flow of the entropy with respect to W (see also the independent work [5] containing related results for Fokker-Planck equations on graphs, as well as [3], where this gradient flow structure has been discovered in the setting of reaction-diffusion systems). In this sense, W takes over the role of the Wasserstein metric, since it is known since the seminal work by Jordan, Kinderlehrer, and Otto that the heat flow on R n is the gradient flow of the entropy [3] (see [, 8, 9,, 3, 36] for variations and generalisations). Convexity along W-geodesics may thus be regarded as a discrete analogue of McCann s displacement convexity [9], which corresponds to convexity along W -geodesics in a continuous setting. Since every pair of probability densities on X can be joined by a W- geodesic, it is possible to define a notion of Ricci curvature in the spirit of Lott Sturm Villani by requiring geodesic convexity of the entropy with respect to the metric W. This possibility has already been indicated in [8]. We shall show that this notion of Ricci curvature shares a number of properties which make the LSV definition so powerful: in particular, it is stable under tensorisation and implies a number of functional inequalities, including a modified logarithmic Sobolev inequality, and a Talagrand-type inequality involving the metric W. Main results. Let us now discuss the contents of this paper in more detail. We work with an irreducible Markov kernel K : X X R + on a finite set X, i.e., we assume that K(x, y) = y X for all x X, and that for every x, y X there exists a sequence {x i } n i= X such that x = x, x n = y and K(x i, x i ) > for all i n. Basic Markov chain theory guarantees the existence of a unique stationary probability measure (also called steady state) π on X, i.e., x X π(x) = and π(y) = x X π(x)k(x, y)

3 RICCI CURVATURE OF FINITE MARKOV CHAINS 3 for all y X. We assume that π is reversible for K, which means that the detailed balance equations hold for x, y X. Let P(X ) := K(x, y)π(x) = K(y, x)π(y) (.) { ρ : X R + x X } π(x)ρ(x) = be the set of probability densities on X. The subset consisting of those probability densities that are strictly positive is denoted by P (X ). We consider the metric W defined for ρ, ρ P(X ) by { } W(ρ, ρ ) := inf (ψ t (x) ψ t (y)) ˆρ t (x, y)k(x, y)π(x) dt, ρ,ψ x,y X where the infimum runs over all sufficiently regular curves ρ : [, ] P(X ) and ψ : [, ] R X satisfying the continuity equation d dt ρ t(x) + (ψ t (y) ψ t (x))ˆρ t (x, y)k(x, y) = x X, y X (.) ρ() = ρ, ρ() = ρ. Here, given ρ P(X ), we write ˆρ(x, y) := ρ(x) p ρ(y) p dp for the logarithmic mean of ρ(x) and ρ(y). The relevance of the logarithmic mean in this setting is due to the identity ρ(x) ρ(y) = ˆρ(x, y)(log ρ(x) log ρ(y)), which somewhat compensates for the lack of a discrete chain rule. The definition of W can be regarded as a discrete analogue of the Benamou Brenier formula [7]. Let us remark that if t ρ t is differentiable at some t and ρ t belongs to P (X ), then the continuity equation (.) is satisfied for some ψ t R X, which is unique up to an additive constant (see [8, Proposition 3.6]). Since the metric W is Riemannian in the interior P (X ), it makes sense to consider gradient flows in (P (X ), W) and it has been proved in [8] that the heat flow associated with the continuous time Markov semigroup P t = e t(k I) is the gradient flow of the entropy H(ρ) = x X π(x)ρ(x) log ρ(x), (.3) with respect to the Riemannian structure determined by W. In this paper we shall show that every pair of densities ρ, ρ P(X ) can be joined by a constant speed geodesic. Therefore the following definition in the spirit of Lott Sturm Villani seems natural. Definition.. We say that K has non-local Ricci curvature bounded from below by κ R if for any constant speed geodesic {ρ t } t [,] in (P(X ), W) we have H(ρ t ) ( t)h(ρ ) + th(ρ ) κ t( t)w(ρ, ρ ).

4 4 MATTHIAS ERBAR AND JAN MAAS In this case, we shall use the notation Ric(K) κ. Remark.. Instead of requiring convexity along all geodesics it will be shown to be equivalent to require that every pair of densities ρ, ρ P(X ) can be joined by a constant speed geodesic along which the entropy is κ- convex. Another equivalent condition would be to impose a lower bound on the Hessian of H in the interior P (X ) (see Theorem 4.5 below for the details). One of the main contributions of this paper is a tensorisation result for non-local Ricci curvature, which we will now describe. For i n, let K i be an irreducible and reversible Markov kernel on a finite set X i, and let π i denote the corresponding invariant probability measure. Let K (i) denote the lift of K i to the product space X = X... X n, defined for x = (x,..., x n ) and y = (y,..., y n ) by { Ki (x K (i) (x, y) = i, y i ), if x j = y j for all j i,, otherwise. For a sequence {α i } i n of nonnegative numbers with n i= α i =, we consider the weighted product chain, determined by the kernel n K α := α i K (i). i= Its reversible probability measure is the product measure π = π π n. Theorem.3 (Tensorisation of Ricci bounds). Assume that Ric(K i ) κ i for i =,..., n. Then we have Ric(K α ) min i α i κ i. Tensorisation results have also been obtained for other notions of Ricci curvature, including the ones by Lott Sturm Villani [4, Proposition 4.6] and Ollivier [34, Proposition 7]. In both cases the proof does not extend to our setting, and completely different ideas are needed here. As a consequence, we obtain a lower bound on the non-local Ricci curvature for (the kernel K n of the simple random walk on) the discrete hypercube {, } n, which turns out to be optimal. Corollary.4. For n we have Ric(K n ) n. The hypercube is a fundamental building block for applications in mathematical physics and theoretical computer science, and the problem of proving displacement convexity on this space has been an open problem that motivated the recent paper by Ollivier and Villani [35], in which a Brunn Minkowski inequality was obtained. Another aspect that we wish to single out at this stage is the fact that Ricci bounds imply a number of functional inequalities, which are natural discrete counterparts to powerful inequalities in a continuous setting. In particular, we obtain discrete counterparts to the results by Bakry Émery [5] and Otto Villani [37].

5 RICCI CURVATURE OF FINITE MARKOV CHAINS 5 To state the results we consider the Dirichlet form E(ϕ, ψ) = ( )( ) ϕ(x) ϕ(y) ψ(x) ψ(y) K(x, y)π(x) x,y X defined for functions ϕ, ψ : X R. Furthermore, we consider the functional I(ρ) = E(ρ, log ρ) defined for ρ P(X ), with the convention that I(ρ) = + if ρ does not belong to P (X ). Its significance here is due to the fact that it is the timederivative of the entropy along the heat flow: d dt H(P tρ) = I(P t ρ). In this sense, I can be regarded as a discrete version of the Fisher information. Theorem.5 (Functional inequalities). Let K be an irreducible and reversible Markov kernel on a finite set X. () If Ric(K) κ for some κ R, then the HWI-inequality H(ρ) W(ρ, ) I(ρ) κ W(ρ, ) (HWI(κ)) holds for all ρ P(X ). () If Ric(K) λ for some λ >, then the modified logarithmic Sobolev inequality H(ρ) λ I(ρ) (MLSI(λ)) holds for all ρ P(X ). (3) If K satisfies (MLSI(λ)) for some λ >, then the modified Talagrand inequality W(ρ, ) λ H(ρ) (T W(λ)) holds for all ρ P(X ). (4) If K satisfies (T W (λ)) for some λ >, then the Poincaré inequality holds for all functions ϕ : X R. ϕ L (X,π) E(ϕ, ϕ) (P(λ)) λ Here, denotes the density of the stationary measure π. The first inequality in Theorem.5 is a discrete counterpart to the HWIinequality from Otto and Villani [37], with the difference that the metric W has been replaced by W. The second result is as a discrete version of the celebrated criterion by Bakry-Émery [5], who proved the corresponding result on Riemannian manifolds. Classically, the Bakry-Émery criterion applies to weighted Riemannian manifolds (M, e V vol M ), and asks for a lower bound on the generalised Ricci curvature given by Ric M + Hess V. As in our setting we allow for general K and π, the potential V is already incorporated in K and π, and our notion of Ricci curvature can be thought of as the analogue of this generalised Ricci curvature. The modified logarithmic Sobolev inequality (MLSI) is motivated by the fact that it yields an explicit rate of exponential decay of the entropy along the heat flow. It has been extensively studied (see, for example, [, 4]),

6 6 MATTHIAS ERBAR AND JAN MAAS along with different discrete logarithmic Sobolev inequalities in the literature (for example, [4, ]). The third part is a discrete counterpart to a famous result by Otto and Villani [37], who showed that the logarithmic Sobolev inequality implies the so-called T -inequality; recall that the T p -inequality is the analogue of T W, in which W is replaced by W p, for p <. These inequalities have been extensively studied in recent years. We refer to [] for a survey and to [4] for a study of the T -inequality in a discrete setting. The modified Talagrand inequality T W that we consider is new. This inequality combines some of the good properties of T and T, as we shall now discuss. Like T, it is weak enough to be applicable in a discrete setting. In fact, we shall prove that T W (λ) holds on the discrete hypercube {, } n with the optimal constant λ = n. By contrast, the T -inequality does not even hold on the two-point space, and it has been an open problem to find an adequate substitute. Like T, and unlike T, T W is strong enough to capture spectral information. In fact, the fourth part in Theorem.5 asserts that it implies a Poincaré inequality with constant λ. Furthermore, we shall show that T W yields good bounds on the sub- Gaussian constant, in the sense that [ E π e t(ϕ E ] ( π[ϕ]) t ) exp (.4) 4λ for all t > and all functions ϕ : X R that are Lipschitz constant with respect to the graph norm. Here, we use the notation E π [ϕ] = x X ϕ(x)π(x). As is well known, this estimate yields the concentration inequality π ( ϕ E π [ϕ] h ) e λh for all h >. The proof of (.4) relies on the fact, proved in Section, that the metric W can be bounded from below by W (with respect to the graph metric), so that T W (λ) implies a T (λ)-inequality, which is known to be equivalent to the sub-gaussian inequality [8]. The proof of Theorem.5 follows the approach by Otto and Villani. On a technical level, the proofs are simpler in the discrete case, since heuristic arguments from Otto and Villani are essentially rigorous proofs in our setting, and no additional PDE arguments are required as in [37]. To summarise, we have the following sequence of implications, for any λ > : { P(λ) Ric(K) λ MLSI(λ) T W (λ) T (λ). Other notions of Ricci curvature. This is of course not the first time that a notion of Ricci curvature has been introduced for discrete spaces, but the notion considered here appears to be the closest in spirit to the one by Lott Sturm Villani. Furthermore, it seems to be the first that yields natural analogues of the results by Bakry Émery and Otto Villani.

7 RICCI CURVATURE OF FINITE MARKOV CHAINS 7 A different notion of Ricci curvature has been introduced by Ollivier [33, 34]. This notion is also based on ideas from optimal transport, and uses the L -Wasserstein metric W, which behaves better in a discrete setting than W. Ollivier s criterion has the advantage of being easy to check in many examples. Furthermore, in some interesting cases it yields functional inequalities with good yet non-optimal constants. Moreover, Ollivier does not assume reversibility, whereas this is strongly used in our approach. It is not completely clear how Ollivier s notion relates to the one by Lott Sturm Villani (see [35] for a discussion). Furthermore, it does not seem to be directly comparable to the concept studied here, as it relies on a metric on the underlying space, which is not the case in our approach. In the setting of graphs, Ollivier s Ricci curvature has been further studied in the recent preprints [6,, 4]. Another approach has been taken by Lin and Yau [6], who defined Ricci curvature in terms of the heat semigroup. Bonciocat and Sturm [] followed a different approach to modify the Lott Sturm Villani criterion, in which they circumvented the lack of midpoints in the L -Wasserstein metric by allowing for approximate midpoints. A Brunn Minkowski inequality in this spirit has been proved on the discrete hypercube by Ollivier and Villani [35]. Organisation of the paper. In Section we collect basic properties of the metric W and formulate an equivalent definition that is more convenient to work with in some situations. Geodesics in the W-metric are studied in Section 3. In particular, it is shown that every pair of densities can be joined by a constant speed geodesic. In Section 4 we present the definition of nonlocal Ricci curvature and give a characterisation in terms of the Hessian of the entropy. Section 5 contains a criterion that allows us to give lower bounds on the Ricci curvature in some basic examples, including the discrete circle and the discrete hypercube. A tensorisation result is contained in Section 6. Finally, we introduce new versions of well-known functional inequalities in Section 7 and prove implications between these and known inequalities. Note added. After essentially finishing this paper, the authors have been informed about the preprint [3], in which geodesic convexity of the entropy for Markov chains has been studied, as well. The results obtained in that paper and this one do not overlap significantly and have been obtained independently. Acknowledgement. The authors are grateful to Nicola Gigli and Karl-Theodor Sturm for stimulating discussions on this paper and related topics. They thank the anonymous referees for detailed comments and valuable suggestions.. The metric W In this section we shall study some basic properties of the metric W. Throughout we shall work with an irreducible and reversible Markov kernel K on a finite set X. The unique steady state will be denoted by π, and we shall write P t := e t(k I), t, to denote the corresponding Markov semigroup. We start by introducing some notation.

8 8 MATTHIAS ERBAR AND JAN MAAS.. Notation. For ϕ R X we consider the discrete gradient ϕ R X X defined by ϕ(x, y) := ϕ(y) ϕ(x). For Ψ R X X we consider the discrete divergence Ψ R X defined by ( Ψ)(x) := (Ψ(x, y) Ψ(y, x))k(x, y) R. With this notation we have y X and the integration by parts formula := = K I, ψ, Ψ π = ψ, Ψ π holds. Here we write, for ϕ, ψ R X and Φ, Ψ R X X, ϕ, ψ π = x X ϕ(x)ψ(x)π(x), Φ, Ψ π = x,y X Φ(x, y)ψ(x, y)k(x, y)π(x). From now on we shall fix a function θ : R + R + R + satisfying the following assumptions: Assumption.. The function θ has the following properties: (A) (Regularity): θ is continuous on R + R + and C on (, ) (, ); (A) (Symmetry): θ(s, t) = θ(t, s) for s, t ; (A3) (Positivity, normalisation): θ(s, t) > for s, t > and θ(, ) = ; (A4) (Zero at the boundary): θ(, t) = for all t ; (A5) (Monotonicity): θ(r, t) θ(s, t) for all r s and t ; (A6) (Positive homogeneity): θ(λs, λt) = λθ(s, t) for λ > and s, t ; (A7) (Concavity): the function θ : R + R + R + is concave. It is easily checked that these assumptions imply that θ is bounded from above by the arithmetic mean: θ(s, t) s + t s, t. (.) In the next result we collect some properties of the function θ, which turn out be very useful in obtaining non-local Ricci curvature bounds. Lemma.. For all s, t, u, v > we have s θ(s, t) + t θ(s, t) = θ(s, t), (.) s θ(u, v) + t θ(u, v) θ(s, t). (.3) Proof. The equality (.) follows immediately from the homogeneity (A6) by noting that the left-hand side equals d dr r= θ(rs, rt). Let us prove (.3). Note that by the concavity (A7) of θ the gradient θ is a monotone operator from R + to R. Hence, for all s, t, x, y > we have ( ) ( ) (s x) θ(s, t) θ(x, y) + (t y) θ(s, t) θ(x, y).

9 RICCI CURVATURE OF FINITE MARKOV CHAINS 9 By the homogeneity (A6) both θ and θ are -homogeneous. now, in particular x = εu, y = εv and letting ε we obtain ( ) ( ) s θ(s, t) θ(u, v) + t θ(s, t) θ(u, v). Taking From this we deduce (.3) by an application of (.). The most important example for our purposes is the logarithmic mean defined by θ(s, t) := s p t p dp = s t log s log t, the latter expression being valid if s, t > and s t. For ρ P(X ) and x, y X we define ˆρ(x, y) = θ(ρ(x), ρ(y)). For a fixed ρ P(X ) it will be useful to consider the Hilbert space G ρ consisting of all (equivalence classes of) functions Ψ : X X R, endowed with the inner product Φ, Ψ ρ := Φ(x, y)ψ(x, y)ˆρ(x, y)k(x, y)π(x). (.4) x,y X Here we identify functions that coincide on the set {(x, y) X X : ˆρ(x, y)k(x, y) > }. The operator can then be considered as a linear operator : L (X ) G ρ, whose negative adjoint is the ρ-divergence operator ( ρ ) : G ρ L (X ) given by ( ρ Ψ)(x) := (Ψ(x, y) Ψ(y, x))ˆρ(x, y)k(x, y). y X.. Equivalent Definitions of the Metric W. We shall now state the definition of the metric W as defined in [8]. Here and in the rest of the paper we will use the shorthand notation A(ρ, ψ) := ψ ρ = (ψ(y) ψ(x)) ˆρ(x, y)k(x, y)π(x) for ρ P(X ) and ψ R X. x,y X Definition.3. For ρ, ρ P(X ) we define { } W( ρ, ρ ) := inf A(ρ t, ψ t ) dt : (ρ, ψ) CE ( ρ, ρ ), where for T >, CE T ( ρ, ρ ) denotes the collection of pairs (ρ, ψ) satisfying the following conditions: (i) ρ : [, T ] R X is C ; (ii) ρ = ρ, ρ T = ρ ; (iii) ρ t P(X ) for all t [, T ] ; (iv) ψ : [, T ] R X is measurable ; (.5) (v) For all x X and all t (, T ) we have ρ t (x) + ( ψt (y) ψ t (x) )ˆρ t (x, y)k(x, y) =. y X

10 MATTHIAS ERBAR AND JAN MAAS Using the notation introduced above, the continuity equation in (v) can be written as ρ t + (ˆρ ψ) =. (.6) Definition.3 is the same as the one in [8], except that slightly different regularity conditions have been imposed on ρ. We shall shortly see that both definitions are equivalent. The following results on the metric W have been proved in [8]. Theorem.4. The following assertions hold. () The space (P(X ), W) is a complete metric space, compatible with the Euclidean topology. () The restriction of W to P (X ) is the Riemannian distance induced by the following Riemannian structure: the tangent space of ρ P (X ) can be identified with the set T ρ := { ψ : ψ R X } by means of the following identification: given a smooth curve ( ε, ε) t ρ t P (X ) with ρ = ρ, there exists a unique element ψ T ρ, such that the continuity equation (.5)(v) holds at t =. The Riemannian metric on T ρ is given by the inner product ϕ, ψ ρ = (ϕ(x) ϕ(y))(ψ(x) ψ(y))ˆρ(x, y)k(x, y)π(x). x,y X (3) If θ is the logarithmic mean, i.e., θ(s, t) = s p t p dp, then the heat flow is the gradient flow of the entropy, in the sense that for any ρ P(X ) and t >, we have ρ t := P t ρ P (X ) and D t ρ t = grad H(ρ t ). (.7) Remark.5. If ρ belongs to P (X ), then the gradient flow equation (.7) also holds for t =. Remark.6. The relevance of the logarithmic mean can be seen as follows. The heat equation ρ t = ρ t = ( ρ t ) can be rewritten as a continuity equation (.6) provided that ψ = ρ ˆρ. On the other hand, an easy computation (see [8, Proposition 4. and Corollary 4.3]) shows that under the identification above, the gradient of the entropy is given by grad W H(ρ) = log ρ. Combining these observations, we infer that the heat flow is the gradient flow of the entropy with respect to W, precisely when ρ ˆρ = log ρ, that is, when θ is the logarithmic mean.

11 RICCI CURVATURE OF FINITE MARKOV CHAINS This argument shows that the same heat flow can also be identified as the gradient flow of the functional F(ρ) = x X f(ρ(x))π(x) for any smooth function f : R R with f >, if one replaces the logarithmic mean by θ(r, s) =. We refer to [8] for the details. r s f (r) f (s) Our next aim is to provide an equivalent formulation of the definition of W, which may seem less intuitive at first sight, but offers several technical advantages. First, the continuity equation becomes linear in V and ρ, which allows us to exploit the concavity of θ. Second, this formulation is more stable so that we can prove existence of minimizers in the class CE ( ρ, ρ ). Similar ideas have already been developed in a continuous setting in [7], where a general class of transportation metrics was constructed based on the usual continuity equation in R n. An important role will be played by the function α : R R + R {+ } defined by α(x, s, t) = The following observation will be useful., θ(s, t) = and x =, x θ(s,t), θ(s, t), +, θ(s, t) = and x. Lemma.7. The function α is lower semicontinuous and convex. Proof. This is easily checked using (A7) and the convexity of the function (x, y) x y on R (, ). Given ρ P(X ) and V R X X we define A (ρ, V ) := α(v (x, y), ρ(x), ρ(y))k(x, y)π(x), and we set where x,y X CE T ( ρ, ρ ) := {(ρ, ψ) : (i ), (ii), (iii), (iv ), (v ) hold }, (i ) ρ : [, T ] R X is continuous ; (iv ) V : [, T ] R X X is locally integrable ; (v ) For all x X we have in the sense of distributions ρ t (x) + ( Vt (x, y) V t (y, x) ) K(x, y) =. y X The continuity equation in (v ) can equivalently be written as ρ t + V =. (.8) As an immediate consequence of Lemma.7 we obtain the following convexity of A. Corollary.8. Let ρ i P(X ) and V i R X X for i =,. For τ [, ] set ρ τ := ( τ)ρ + τρ and V τ := ( τ)v + τv. Then we have A (ρ τ, V τ ) ( τ)a (ρ, V ) + τa (ρ, V ). Now we have the following reformulation of Definition.3.

12 MATTHIAS ERBAR AND JAN MAAS Lemma.9. For ρ, ρ P(X ) we have { } W( ρ, ρ ) = inf A (ρ t, V t ) dt : (ρ, V ) CE ( ρ, ρ ). Furthermore, if ρ, ρ P (X ), condition (iv) in (.5) can be reinforced into: ψ : [, T ] R X is C. Proof. The inequality follows easily by noting that the infimum is taken over a larger set. Indeed, given a pair (ρ, ψ) CE ( ρ, ρ ) we obtain a pair (ρ, V ) CE ( ρ, ρ ) by setting V t (x, y) = ψ t (x, y)ˆρ t (x, y) and we have A (ρ t, V t ) = A(ρ t, ψ t ). To show the opposite inequality, we fix an arbitrary pair (ρ, V ) CE ( ρ, ρ ). It is sufficient to show that for every ε > there exists a pair (ρ ε, ψ ε ) CE ( ρ, ρ ) such that A(ρ ε t, ψ ε t ) dt A (ρ t, V t ) dt + ε. For this purpose we first regularise (ρ, V ) by a mollification argument. We thus define ( ρ, Ṽ ) : [ ε, + ε] P(X ) RX X by (ρ(), ), t [ ε, ε), ( ρ t, Ṽt) = (ρ( t ε ε ), ε V ( t ε ε )), t [ε, ε), (ρ(), ), t [ ε, + ε], and take a nonnegative smooth function η : R R + which vanishes outside of [ ε, ε], is strictly positive on ( ε, ε) and satisfies η(s) ds =. For t [, ] we define ρ ε t = η(s) ρ t+s ds, Vt ε = η(s)ṽt+s ds. Now t ρ ε t is C and using the continuity of ρ it is easy to check that (ρ ε, V ε ) CE ( ρ, ρ ). Moreover, using the convexity from Corollary.8 we can estimate A (ρ ε t, Vt ε ) dt η(s)a ( ρ t+s, Ṽt+s) ds dt +ε ε A ( ρ t, Ṽt) dt = ε A (ρ t, V t ) dt. To proceed further, we may assume without loss of generality that V (x, y) = whenever K(x, y) =. The fact that A (ρ t, V t ) dt is finite implies that the set {t : ˆρ t (x, y) = and V t (x, y) } is negligible for all x, y X. Taking properties (A3) and (A4) of the function θ into account, this implies that for the convolved quantities the corresponding set {t : ˆρ ε t(x, y) = and Vt ε (x, y) } is empty for all x, y X. Hence there exists a measurable function Ψ ε : [, ] R X X satisfying ρ ε t V ε t (x, y) = Ψ ε t(x, y)ˆρ ε t(x, y) for all x, y X and all t [, ]. (.9) It remains to find a function ψ ε : [, ] R X such that ρ ε t Ψ ε t = ψt ε. Let P ρ denote the orthogonal projection in G ρ onto the range

13 RICCI CURVATURE OF FINITE MARKOV CHAINS 3 of. Then there exists a measurable function ψ ε : [, ] R X such that P ρ ε t Ψ ε t = ψ ε t. The orthogonal decomposition G ρ ε t = Ran( ) Ker( ρ ε t ) (.) implies that ρ ε t Ψ ε t = ρ ε t ψt ε, hence (ρ ε, ψ ε ) CE ( ρ o, ρ ). Using the decomposition (.) once more, we infer that ψt ε, ψt ε ρ ε t Ψ ε t, Ψ ε t ρ ε t. This implies A(ρ ε t, ψt ε ) A (ρ ε t, Vt ε ) and finishes the proof of the first assertion. If ρ and ρ belong to P (X ), one can follow the argument in [8, Lemma 3.3] and construct a curve ( ρ, V ) CE ( ρ, ρ ) such that ρ t P (X ) for t [, ] and A ( ρ t, V t ) dt A (ρ t, V t ) dt + ε. Then one can apply the argument above. In this case, ρ ε t(x) > for all x X and t [, ], and therefore the function Ψ ε : [, ] R X X is C. Furthermore, since the orthogonal projection P ρ depends smoothly on ρ P (X ), the function ψ ε : [, ] R X is smooth as well. Remark.. In [8] the metric W has been defined as in Definition.3, with the difference that (i) in (.8) was replaced by ρ : [, T ] P(X ) is piecewise C. Therefore Lemma.9 shows, in particular, that Definition.3 coincides with the original definition of W from [8]..3. Basic properties of W. As an application of Lemma.9 we shall prove the following convexity result, which is a discrete counterpart of the well-known fact that the squared L -Wasserstein distance over Euclidean space is convex with respect to linear interpolation (see, for example, [7, Theorem 5.]). Proposition. (Convexity of the squared distance). For i, j =,, let ρ j i P(X ), and for τ [, ] set ρτ i := ( τ)ρ i + τρ i. Then W(ρ τ, ρ τ ) ( τ)w(ρ, ρ ) + τw(ρ, ρ ). Proof. Let ε >. For j =, we may take a pair (ρ j, V j ) CE (ρ j, ρj ) with A (ρ j t, V j t ) dt W (ρ j, ρj ) + ε in view of Lemma.9. For τ [, ] we set ρ τ t := ( τ)ρ t + τρ t, V τ t := ( τ)v t + τv t. It then follows that (ρ τ, V τ ) CE (ρ τ, ρτ ), hence by Corollary.8, W(ρ τ, ρ τ ) ( τ) A (ρ τ t, Vt τ ) dt A (ρ t, V t ) dt + τ = ( τ)w(ρ, ρ ) + τw(ρ, ρ ) + ε. Since ε > is arbitrary, this completes the proof. A (ρ t, V t ) dt

14 4 MATTHIAS ERBAR AND JAN MAAS In this section we compare W to some commonly used metrics. A first result of this type (see [8, Lemma 3.]) gives a lower bound on W in terms of the total variation metric d T V (ρ, ρ ) = x X π(x) ρ (x) ρ (x). Here, more generally, we shall compare W to various Wasserstein distances. Given a metric d on X and p <, recall that the L p -Wasserstein metric W p,d on P(X ) is defined by {( W p,d (ρ, ρ ) := inf x,y X ) } d(x, y) p p q(x, y) q Γ(ρ, ρ ), (.) where Γ(ρ, ρ ) denotes the set of all couplings between ρ and ρ, i.e., { Γ(ρ, ρ ) := q : X X R + q(x, y) = ρ (x)π(x), y X x X } q(x, y) = ρ (y)π(y). It is well known (see, for example, [43, Theorem 4.]) that the infimum in (.) is attained; as usual we shall denote the collection of minimizers by Γ o (ρ, ρ ). In our setting there are various metrics on X that are natural to consider. In particular, the graph distance d g with respect to the graph structure on X induced by K (i.e., {x, y} is an edge iff K(x, y) > ). the metric d W, that is, the restriction of W from P(X ) to X under the identification of points in X with the corresponding Dirac masses: ( {x} d W (x, y) := W π(x), ) {y}. π(y) The induced L p -Wasserstein distances will be denoted by W p,g and W p,w respectively. We shall now prove lower and upper bounds for the metric W in terms of suitable Wasserstein metrics. We start with the lower bounds. Let us remark that, unlike most other results in this paper, the second inequality in the following result relies on the normalisation y X K(x, y) =. Proposition. (Lower bounds for W). For all probability densities ρ, ρ P(X ) we have d T V (ρ, ρ ) W,g (ρ, ρ ) W(ρ, ρ ). (.) Proof. Note that d tr d g, where d tr (x, y) = x y denotes the trivial distance. Therefore, the first bound follows from the fact that d T V is the L -Wasserstein distance induced by d tr (see [4, Theorem.4]).

15 RICCI CURVATURE OF FINITE MARKOV CHAINS 5 In order to prove the second bound, we fix ε >, take ρ, ρ P(X ) and (ρ, ψ) CE ( ρ, ρ ) with ( ) A(ρ t, ψ t ) dt W( ρ, ρ ) + ε. Using the continuity equation from (.5) we obtain for any ϕ : X R, ϕ(x)(ρ (x) ρ (x))π(x) x X = = ϕ(x) ρ t (x)π(x) dt x X x,y X = ϕ, ψ t ρt dt ( ϕ(x) ( ψ t (x) ψ t (y) )ˆρ t (x, y)k(x, y)π(x) dt ) / ( ) / ϕ ρ t dt ψ t ρ t dt ( / = ϕ ρ t dt) (W( ρ, ρ ) + ε). Let [ϕ] Lip denote the Lipschitz constant of ϕ with respect to the graph distance d g, i.e., ϕ(x) ϕ(y) [ϕ] Lip := sup. x y d g (x, y) Applying the inequality (.) and using the fact that d g (x, y) = if x y and K(x, y) >, we infer that ϕ ρ t = ( ) K(x, ϕ(x) ϕ(y) y)ˆρt (x, y)π(x) x,y X 4 [ϕ] Lip x,y X K(x, y) ( ρ t (x) + ρ t (y) ) π(x) = [ϕ] Lip ρ t (x)π(x) K(x, y) y X = [ϕ] Lip. x X The Kantorovich Rubinstein Theorem (see, for example, [4, Theorem.4]) yields W,g ( ρ, ρ ) = sup ϕ(x)( ρ (x) ρ (x))π(x) W( ρ, ρ ) + ε, x X ϕ:[ϕ] Lip which completes the proof, since ε > is arbitrary. Before stating the upper bounds, we provide a simple relation between d g and d W.

16 6 MATTHIAS ERBAR AND JAN MAAS Lemma.3. For x, y X we have where c = d W (x, y) c k d g (x, y), dr < and k = min K(x, y). θ( r, + r) (x,y) : K(x,y)> If θ is the logarithmic mean, then c.56. Proof. Let {x i } n i= be a sequence in X with x = x, x n = y and K(x i, x i+ ) > for all i. We shall use the fact, proved in [8, Theorem.4], that the W-distance between two Dirac measures on a two-point space {a, b} with transition probabilities K(a, b) = K(b, a) = p is equal to c p. The concavity of θ readily implies that c is finite. Furthermore, it follows from [8, Lemma 3.4] and its proof, that for any pair x, y X with K(x, y) >, one has ( {x} W π(x), ) {y} c π(y) max{π(x), π(y)} K(x, y)π(x) Using the triangle inequality for W we obtain c k. ( {x} d W (x, y) = W π(x), ) n ( {y} {xi } W π(y) π(x i ), ) {x i+} nc, π(x i+ ) k i= hence the result follows by taking the infimum over all such sequences {x i } n i=. Now we turn to upper bounds for W in terms of L -Wasserstein distances. Proposition.4 (Upper bounds for W). For all probability densities ρ, ρ P(X ) we have W(ρ, ρ ) W,W (ρ, ρ ) where c and k are as in Lemma.3. c k W,g (ρ, ρ ), (.3) Proof. We shall prove the first bound, the second one being an immediate consequence of Lemma.3. For this purpose, we fix ρ, ρ P(X ) and take q Γ o ( ρ, ρ ). For all u, v X, take a curve (ρ u,v, V u,v ) CE ( {u} π(u), ) {v} π(v) with A (ρ u,v t, V u,v t ) dt d W (u, v) + ε, and consider the convex combination of these curves, weighted according to the optimal plan q, i.e., ρ t := u,v X q(u, v)ρ u,v t, V t := u,v X q(u, v)v u,v t.

17 RICCI CURVATURE OF FINITE MARKOV CHAINS 7 It then follows that the resulting curve (ρ, V ) belongs to CE ( ρ, ρ ). Using the convexity result from Lemma.7 we infer that W( ρ, ρ ) A (ρ t, V t ) dt u,v X u,v X q(u, v) A (ρ u,v t q(u, v)(d W (u, v) + ε) = W,W ( ρ, ρ ) + ε., V u,v t ) dt which implies the result. 3. Geodesics In this section we show that the metric space (P(X ), W) is a geodesic space, in the sense that any two densities ρ, ρ P(X ) can be connected by a (constant speed) geodesic, that is, a curve γ : [, ] P(X ) satisfying W(γ s, γ t ) = s t W(γ, γ ) for all s, t. Let us first give an equivalent characterisation of the infimum in Lemma.9, which is invariant under reparametrisation. Lemma 3.. For any T > and ρ, ρ P(X ) we have { T } W( ρ, ρ ) = inf A (ρ t, V t )dt : (ρ, V ) CE T ( ρ, ρ ). (3.) Proof. Taking Lemma.9 into account, this follows from a standard reparametrisation argument. See [, Lemma..4] or [7, Theorem 5.4] for details in similar situations. Theorem 3.. For all ρ, ρ P(X ) the infimum in Lemma.9 is attained by a pair (ρ, V ) CE ( ρ, ρ ) satisfying A (ρ t, V t ) = W( ρ, ρ ) for a.e. t [, ]. In particular, the curve (ρ t ) t [,] is a constant speed geodesic. Proof. We will show existence of a minimizing curve by a direct argument. Let (ρ n, V n ) CE ( ρ, ρ ) be a minimizing sequence. Thus we can assume that sup n A (ρ n t, V n t ) dt < C for some finite constant C. Without loss of generality we assume that Vt n (x, y) = when K(x, y) =. For x, y X, define the sequence of signed Borel measures νx,y n on [, ] by νx,y(dt) n := Vt n (x, y) dt. For every Borel set B [, ] we can give the following bound on the total variation of these measures: νx,y (B) n Vt n (x, y) dt C α(vt n(x, y), ρn t (x), ρn t (y)) dt, B B

18 8 MATTHIAS ERBAR AND JAN MAAS where we used the fact that ρ(x) max{π(z) : z X } =: C < for ρ P(X ). Using Hölder s inequality we obtain x,y X νx,y (B)K(x, n y)π(x) ( ) C Leb(B) A (ρ n t, Vt n ) dt CC Leb(B). (3.) In particular, the total variation of the measures νx,y n is bounded uniformly in n. Hence we can extract a subsequence (still indexed by n) such that for all x, y X the measures νx,y n converge weakly* to some finite signed Borel measure ν x,y. The estimate (3.) also shows that ν x,y is absolutely continuous with respect to the Lebesgue measure. Thus there exists V : [, ] R X X such that ν x,y (dt) := V t (x, y) dt. We claim that, along the same subsequence, ρ n converges pointwise to a function ρ : [, ] P(X ). Indeed, using the continuity of t ρ n t one derives from the continuity equation (v ) in (.8) that for s [, ] and every x X, ρ n s ρ n = s y X (V n t (y, x) V n t (x, y))k(x, y) dt. (3.3) The weak* convergence of νx,y n implies (see [, Prop. 5..]) the convergence of the right-hand side of (3.3). Since ρ n = ρ for all n, this yields the desired convergence of ρ n s for all s, and one easily checks that (ρ, V ) CE (ρ, ρ ). The weak* convergence of νx,y n further implies that the measures ρ n t (x)dt converge weakly* to ρ t (x)dt. Applying a general result on the lowersemicontinuity of integral functionals (see [3, Thm ]) and taking into account Lemma.7, we obtain A (ρ t, V t ) dt lim inf n A (ρ n t, V n t ) dt = W( ρ, ρ ). Hence the pair (ρ, V ) is a minimizer of the variational problem in the definition of W. Finally, Lemma 3. yields A (ρ t, V t ) dt W( ρ, ρ ) = ( ) A (ρ t, V t ) dt which implies that A (ρ t, V t ) = W ( ρ, ρ ) for a.e. t [, ]. The fact that (ρ t ) t is a constant speed geodesic follows now by another application of Lemma 3.. We shall now give a characterisation of absolutely continuous curves in the metric space (P(X ), W) and relate their length to their minimal action. First we recall some notions from the theory of analysis in metric spaces. A curve (ρ t ) t [,T ] in P(X ) is called absolutely continuous w.r.t. W if there exists m L (, T ) such that W(ρ s, ρ t ) t s m(r) dr for all s t T.,

19 RICCI CURVATURE OF FINITE MARKOV CHAINS 9 If (ρ t ) is absolutely continuous, then its metric derivative ρ t := lim h W(ρ t+h, ρ t ) h exists for a.e. t [, T ] and satisfies ρ t m(t) a.e. (see [, Theorem..]). Proposition 3.3 (Metric velocity). A curve (ρ t ) t [,T ] is absolutely continuous with respect to W if and only if there exists a measurable function V : [, T ] R X X such that (ρ, V ) CE T (ρ, ρ T ) and T A (ρ t, V t )dt <. In this case we have ρ t A (ρ t, V t ) for a.e. t [, T ] and there exists an aalmost everywhere uniquely defined function Ṽ : [, ] RX X such that (ρ, Ṽ ) CE T (ρ, ρ T ) and ρ t = A (ρ t, Ṽt) for a.e. t [, T ]. Proof. The proof follows from the very same arguments as in [7, Thm. 5.7]. To construct the velocity field Ṽ, the curve ρ is approximated by curves (ρ n, V n ) which are piecewise minimizing. The velocity field Ṽ is then defined as a subsequential limit of the velocity fields V n. In our case, existence of this limit is guaranteed by a compactness argument similar to the one in the proof of Theorem 3.. For later use we state an explicit formula for the geodesic equations in P (X ) from [8, Proposition 3.4]. Since the interior P (X ) of P(X ) is Riemannian by Theorem.4, local existence and uniqueness of geodesics is guaranteed by standard Riemannian geometry. Proposition 3.4. Let ρ P (X ) and ψ R X. On a sufficiently small time interval around, the unique constant speed geodesic with ρ = ρ and initial tangent vector ψ = ψ satisfies the following equations: t ρ t (x) + t (y) ψ t (x))ˆρ t (x, y)k(x, y) =, y X(ψ t ψ t (x) + ( ψt (x) ψ t (y) ) (3.4) θ(ρ t (x), ρ t (y))k(x, y) =. y X 4. Ricci curvature In this section we initiate the study of a notion of Ricci curvature lower boundedness in the spirit of Lott, Sturm, and Villani [7, 4]. Furthermore, we present a characterisation, which we shall use to prove Ricci bounds in concrete examples. As before, we fix an irreducible and reversible Markov kernel K on a finite set X with steady state π. The associated Markov semigroup shall be denoted by (P t ) t. Assumption 4.. Throughout the remainder of the paper we assume that θ is the logarithmic mean. We are now ready to state the definition, which has already been given in [8, Definition.3].

20 MATTHIAS ERBAR AND JAN MAAS Definition 4.. We say that K has non-local Ricci curvature bounded from below by κ R and write Ric(K) κ, if the following holds: for every constant speed geodesic (ρ t ) t [,] in (P(X ), W) we have H(ρ t ) ( t)h(ρ ) + th(ρ ) κ t( t)w(ρ, ρ ). (4.) An important role in our analysis is played by the quantity B(ρ, ψ), which is defined for ρ P (X ) and ψ R X by B(ρ, ψ) := where = 4 ˆ ρ ψ, ψ x,y,z X π ˆρ ψ, ψ π ( ) ( ψ(x) ψ(y) θ ( ρ(x), ρ(y) )( ρ(z) ρ(x) ) K(x, z) + θ ( ρ(x), ρ(y) )( ρ(z) ρ(y) ) ) K(y, z) K(x, y)π(x) (K(x, z) ( ψ(z) ψ(x) ) K(y, z) ( ψ(z) ψ(y) )) x,y,z X ( ψ(x) ψ(y) )ˆρ(x, y)k(x, y)π(x), ˆ ρ(x, y) := θ(ρ(x), ρ(y)) ρ(x) + θ(ρ(x), ρ(y)) ρ(y). The significance of B(ρ, ψ) is mainly due to the following result: Proposition 4.3. For ρ P (X ) and ψ R X we have Hess H(ρ) ψ, ψ = B(ρ, ψ). Proof. Take (ρ, ψ) satisfying the geodesic equations (3.4), so that Hess H(ρt ) ψ t, ψ t ρ t = d dt H(ρ t). Using the continuity equation we obtain Furthermore, d dt H(ρ t) = + log ρ t, (ˆρ t ψ t ) π = log ρ t, ˆρ t ψ t π = ρ t, ψ t π. d dt H(ρ t) = t ρ t, ψ t π + ρ t, t ψ t π = t ρ t, ψ t π ρ t, t ψ t π. Using the continuity equation we obtain t ρ t, ψ t π = (ˆρ ψ t ), ψ t = ˆρ t ψ t, ψ t ρ π π = ψ t, ψ t ρ t. (4.)

21 RICCI CURVATURE OF FINITE MARKOV CHAINS Furthermore, applying the geodesic equations (3.4) and the detailed balance equations (.), we infer that ρt, t ψ t = = 4 π x,y,z X x,y,z X ( ψt (x) ψ t (y) ) θ ( ρ t (x), ρ t (y) ) ( ρ t (z) ρ t (x) ) K(x, y)k(x, z)π(x) ( ψt (x) ψ t (y) ) ( θ ( ρ t (x), ρ t (y) )( ρ t (z) ρ t (x) ) K(x, z) + θ ( ρ t (x), ρ t (y) )( ρ t (z) ρ t (y) ) ) K(y, z) K(x, y)π(x) = ˆ ρt ψ t, ψ t π. Combining the latter three identities, we arrive at d dt H(ρ t) = ψ t, ψ t ρ t + ˆ ρt ψ t, ψ t π, which is the desired identity. Our next aim is to show that κ-convexity of H along geodesics is equivalent to a lower bound of the Hessian of H in P (X ). Since the Riemannian metric on (P(X ), W) degenerates at the boundary, this is not an obvious result. In particular, in order to prove the implication (4) (3) below we cannot directly apply the equivalence between the so-called EVI (4.4) and the usual gradient flow equation, which holds on complete Riemannian manifolds (see, for example, [43, Proposition 3.]). Therefore, we take a different approach, based on an argument by Daneri and Savaré [6], which avoids delicate regularity issues for geodesics. An additional benefit of this approach is that we expect it to apply in a more general setting where the underlying space X is infinite, and finite-dimensional Riemannian techniques do not apply at all. Remark 4.4. The quantity B(ρ, ψ) arises naturally in the Eulerian approach to the Wasserstein metric, as developed in [6, 38]. In fact, in a crucial argument from [6], the authors consider a certain two-parameter family of measures (ρ s t) and functions (ψt s ) on a Riemannian manifold M, and show that s H(ρ s t) + t ψt s dρ s t = B(ρ s t, ψt s ), (4.3) where B(ρ, ψ) := M M ( ( ψ ) ψ, ψ ) dρ. Since Bochner s formula asserts that B(ρ, ψ) := D ψ + Ric( ψ, ψ) dρ, M one obtains a lower bound on B if the Ricci curvature is bounded from below. The lower bound on B can be used to prove an evolution variational inequality, which in turn yields convexity of the entropy along W -geodesics.

22 MATTHIAS ERBAR AND JAN MAAS In our setting, the quantity B(ρ, ψ) can be regarded as a discrete analogue of B(ρ, ψ). Therefore the inequality B(ρ, ψ) κa(ρ, ψ) could be interpreted as a one-sided Bochner inequality, which allows us to adapt the strategy from [6] to the discrete setting. In the following result and the rest of the paper we shall use the notation d + f(t + h) f(t) f(t) = lim sup. dt h h Theorem 4.5. Let κ R. For an irreducible and reversible Markov kernel (X, K) the following assertions are equivalent: () Ric(K) κ ; () For all ρ, ν P(X ), the following evolution variational inequality holds for all t : d + dt W (P t ρ, ν) + κ W (P t ρ, ν) H(ν) H(P t ρ) ; (4.4) (3) For all ρ, ν P (X ), (4.4) holds for all t ; (4) For all ρ P (X ) and ψ R X we have (5) For all ρ P (X ) we have B(ρ, ψ) κa(ρ, ψ). Hess H(ρ) κ ; (6) For all ρ, ρ P (X ) there exists a constant speed geodesic (ρ t ) t [,] satisfying ρ = ρ, ρ = ρ, and (4.). Proof. (3) () : This is a special case of [6, Theorem 3.3]. () () : This follows by applying [6, Theorem 3.] to the metric space (P(X ), W) and the functional H. () (6) : This is clear in view of Theorem 3.. (6) (5) : Take ρ P (X ) and ψ R X and consider the unique solution (ρ t, ψ t ) t ( ε,ε) to the geodesic equations with ρ = ρ and ψ = ψ on a sufficiently small time interval around. Using the local uniqueness of geodesics and (6), we infer that Hess H(ρ)( ψ) = d dt t= H(ρ t ) κ ψ ρ (see, for example, the implication (ii) (i) in [43, Proposition 6.]). (5) (4) : This follows from Proposition 4.3. (4) (3) : We follow [6]. In view of Lemma.9 we can find a smooth curve (ρ, ψ ) CE (ν, ρ) satisfying A(ρ s, ψ s ) ds < W(ρ, ν) + ε. (4.5) Note in particular that s ρ s and s ψ s are sufficiently regular to apply Lemma 4.6 below. Using the notation from this lemma, we infer that ta(ρ s t, ψ s t ) + s H(ρ s t) = sb(ρ s t, ψ s t ).

23 RICCI CURVATURE OF FINITE MARKOV CHAINS 3 Using the assumption that B κa we infer that ( ) ( ) t e κst A(ρ s t, ψt s ) + s e κst H(ρ s t) κte κst H(ρ s t). Integration with respect to t [, h] and s [, ] yields ( ) e κsh A(ρ s h, ψs h ) A(ρs, ψ) s ds h ( ) h + e κt H(ρ t ) H(ρ t ) dt κ te κst H(ρ s t) dt ds. Arguing as in [6, Lemma 5.] we infer that e κsh A(ρ s h, ψs h ) ds m(κh)w (P h ρ, ν), where m(κ) = κeκ sinh(κ). Using (4.5) together with the fact that the entropy decreases along the heat flow, we infer that m(κh) W (P h ρ, ν) W (ρ, ν) ε + E κ (h)h(p h ρ) hh(ν) κ h te κst H(ρ s t) dt ds, where E κ (h) := h eκt dt. Since H is bounded, it follows that Furthermore, lim h h h te κst H(ρ s t) dt ds =. ( ) lim E κ (h)h(p h ρ) hh(ν) = H(ρ) H(ν). h h Since ε > is arbitrary, (4.6) implies that d + ( ) m(κh) dh W (P h ρ, ν) + H(ρ) H(ν). h= Taking into account that d + ( ) m(κh) dh W (P h ρ, ν) = κ h= W (ρ, ν) + d + dh W (P h ρ, ν), h= we obtain (4.4) for t =, which clearly implies (4.4) for all t. (4.6) The following result, which is used in the proof of Theorem 4.5, is a discrete analogue of (4.3) and the proof proceeds along the lines of [6, Lemma 4.3]. Since the details are slightly different in the discrete setting, we present a proof for the convenience of the reader. Lemma 4.6. Let {ρ s } s [,] be a smooth curve in P(X ). For each t, set ρ s t := e st ρ s, and let {ψ s t } s [,] be a smooth curve in R X satisfying the continuity equation s ρ s t + (ˆρ s t ψ s t ) =, s [, ].

24 4 MATTHIAS ERBAR AND JAN MAAS Then the identity ta(ρ s t, ψt s ) + s H(ρ s t) = sb(ρ s t, ψt s ) holds for every s [, ] and t. Proof. First of all, we have s H(ρ s t) = + log ρ s t, s ρ s t Furthermore, = + log ρ s t, (ˆρ s t ψ) π = log ρ s t, ˆρ s t ψt s π = ρ s t, ψt s π = ψt s, ρ s t π. ta(ρ s t, ψt s ) = ˆρ s t t ψt s, ψt s π + t ˆρ s t ψt s, ψt s =: I + I. In order to simplify I we claim that π π (4.7) (( t ˆρ s t) ψ s t ) (ˆρ s t t ψ s t ) = ρ s t s ( (ˆρ s t ψ s t ) ), (4.8) t ˆρ s t = s ˆ ρ s t. (4.9) To show (4.8), note that the left-hand side equals t s ρ s t, while the righthand side equals s t ρ s t. The identity (4.9) follows from a straightforward calculation. Integrating by parts repeatedly and using (4.7), (4.8) and (4.9), we obtain I = ψt s, (ˆρ s t t ψt s ) π = ψt s, ρ s t π s ψt s, ( (ˆρ s t ψt s ) ) π + ψt s, (( t ˆρ s t) ψt s ) = s H(ρ s t) + s ˆρ s t ψt s, ψt s π s s ˆ ρ t ψt s, ψt s π. Taking into account that I = s ˆ ρ s t ψt s, ψt s π, the result follows by summing the expressions for I and I. The evolution variational inequality (4.4) has been extensively studied in the theory of gradient flows in metric spaces []. It readily implies a number of interesting properties for the associated gradient flow (see, for example, [6, Section 3]). Among them we single out the following κ-contractivity property. Proposition 4.7 (κ-contractivity of the heat flow). Let (X, K) be an irreducible and reversible Markov kernel satisfying Ric(K) κ for some κ R. Then the associated continuous time Markov semigroup (P t ) t satisfies for all ρ, σ P(X ) and t. W(P t ρ, P t σ) e κt W(ρ, σ) Proof. This follows by applying [6, Proposition 3.] to the functional H on the metric space (P(X ), W). π

25 RICCI CURVATURE OF FINITE MARKOV CHAINS 5 5. Examples In this section we give explicit lower bounds on the non-local Ricci curvature in several examples. Moreover, we present a simple criterion (see Proposition 5.4) for proving non-local Ricci curvature bounds. Although the assumptions seem restrictive, the criterion allows us to obtain the sharp Ricci bound for the discrete hypercube. Moreover, it can be combined with the tensorisation result from Section 6 in order to prove Ricci bounds in other nontrivial situations. To get started let us consider a particularly simple example. Example 5. (The complete graph). Let K n denote the complete graph on n vertices and let K n be the simple random walk on K n given by the transition kernel K(x, y) = n for all x, y Kn. Note that in this case π is the uniform measure. We will show that Ric(K n ) + n. In view of Theorem 4.5 we have to show B(ρ, ψ) ( + n )A(ρ, ψ) for all ρ P (X ) and ψ R X. Recall the definition (4.) of the quantity B. We calculate explicitly: ˆρ ψ, ψ π = [ ] n 3 ˆρ(x, y) ψ(y, x) ψ(x, z) ψ(y, z) x,y,z X = n ˆρ(x, y) ( ψ(x, y) ) = A(ρ, ψ). x,y With the notation ˆρ i (x, y) = i θ(ρ(x), ρ(y)) and using equation (.) we obtain further ˆ ρ ψ, ψ π = [ ( ) ψ(x, y) n 3 ˆρ (x, y)(ρ(z) ρ(x)) x,y,z ] + ˆρ (x, y)(ρ(z) ρ(y)) = A(ρ, ψ) + [ ( ) ψ(x, y) n 3 ˆρ (x, y)ρ(z) x,y,z ] + ˆρ (x, y)ρ(z). Keeping only the terms with z = x (resp. z = y) in the last sum and using (.) again, we see ˆ ρ ψ, ψ π ( n ) A(ρ, ψ). Summing up, we obtain B ( ( n ) + )A, which yields the claim. For the rest of this section we let K be an irreducible and reversible Markov kernel on a finite set X. In order to state the criterion and to perform calculations, it will be convenient to write a Markov chain in terms of allowed moves rather than jumps from point to point. Let G be a set of maps from X to itself (the allowed moves) and consider a function c : X G R + (representing the jump rates).

26 6 MATTHIAS ERBAR AND JAN MAAS Definition 5.. We call the pair (G, c) a mapping representation of K if the following properties hold: () The generator = K I can be written in the form ψ(x) = δ G δ ψ(x)c(x, δ), (5.) where δ ψ(x) = ψ(δx) ψ(x). () For every δ G there exists a unique δ G satisfying δ (δ(x)) = x for all x with c(x, δ) >. (3) For every F : X G R we have F (x, δ)c(x, δ)π(x) = F (δx, δ )c(x, δ)π(x). (5.) x X,δ G x X,δ G Remark 5.3. This definition is close in spirit to the recent work [4], where Γ -type calculations have been performed in order to prove strict convexity of the entropy along the heat flow in a discrete setting. Here, we essentially compute the second derivatives of the entropy along W-geodesics. Since the geodesic equations are more complicated than the heat equation, the expressions that we need to work with are somewhat more involved. Every irreducible, reversible Markov chain has a mapping representation. In fact, an explicit mapping representation can be obtained as follows. For x, y X consider the bijection t {x,y} : X X that interchanges x and y and keeps all other points fixed. Then let G be the set of all these transpositions and set c(x, t {x,y} ) = K(x, y) and c(x, t {y,z} ) = for x / {y, z}. Then (G, c) defines a mapping representation. However, in examples it is often more natural to work with a different mapping representation involving a smaller set G, as we shall see below. It will be useful to formulate the expressions for A and B in this formalism. For this purpose, we note that (5.) implies that F (x, δx)c(x, δ) y X F (x, y)k(x, y) = δ G for any F : X X R vanishing on the diagonal. As a consequence we obtain A(ρ, ψ) = ( δ ψ(x) ) ˆρ(x, δx)c(x, δ)π(x) (5.3) x X,δ G and ˆρ ψ, ψ π = x X δ,η G [ δ ψ(x) η ψ(δx)c(δx, η) ] η ψ(x)c(x, η) ˆρ(x, δx)c(x, δ)π(x). (5.4)