CONVERGENCE TO GLOBAL EQUILIBRIUM FOR FOKKER-PLANCK EQUATIONS ON A GRAPH AND TALAGRAND-TYPE INEQUALITIES

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1 CONVERGENCE TO GLOBAL EQUILIBRIUM FOR FOKKER-PLANCK EQUATIONS ON A GRAPH AND TALAGRAND-TYPE INEQUALITIES RUI CHE, WEN HUANG, YAO LI, AND PRASAD TETALI Abstract. In recent work, Chow, Huang, Li and Zhou [6] introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When N 2 is the number of vertices of the graph, they show that the corresponding Fokker- Planck equation is a system of N nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, which is a Gibbs distribution. In this paper we study the speed of convergence towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove that the convergence is indeed exponential. The rate as measured by the decay of the L 2 norm can be bound in terms of the spectral gap of the Laplacian of the graph, and as measured by the decay of relative) entropy be bound using the modified logarithmic Sobolev constant of the graph. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [6]. The first one is a local inequality, while the second is a global inequality with respect to the lower bound metric from [6]. 1. Introduction As the stochastic differential equation becomes one of the primary and highly effective tools in many practical problems arising in diverse fields such as finance, physics, chemistry and biology [8, 19, 23], there are considerable efforts in understanding the properties of the classical Fokker-Planck equation that describes the time evolution of the probability distribution of a stochastic process. At the same time, the free energy functional, which is defined on the space of probability distributions, as a linear combination of terms involving a potential and an entropy, has widely been used in various subjects; it typically means different things in different contexts. For example, the notion of free energy in thermodynamics is related to the maximal amount of work that can be extracted from a system. The concept of free energy is also used in other fields, such as statistical mechanics, probability particularly in the context of Markov Random Fields), biology, chemistry, and image processing; see e.g., [14, 26, 3]. Since the seminal work of Jordan, Kinderlehrer and Otto [1, 17], it is well known that a Fokker-Planck equation is the gradient flow of the free energy functional on a Riemannian manifold that is defined by a space of probability distributions with a 2-Wasserstein 2 Mathematics Subject Classification. Primary: 37H1,6J27,6J6. Key words and phrases. Fokker-Planck equation, Gibbs density, graph, Talagrand inequality. The second author is supported by NSFC No ). The work of the last author is supported in part by NSF DMS

2 2 R. CHE, W. HUANG, Y. LI, AND P. TETALI metric on it. This discovery has been the starting point for many developments relating the free energy, Fokker-Planck equation, an abstract notion of a Ricci curvature and optimal transport theory in continuous spaces. We refer to the monographs [1, 28, 29] for an overview. Recently, a synthetic theory of Ricci curvature in length spaces has been developed by Lott-Sturm-Villani [11, 24, 25], which reveals the fundamental relationship between entropy and Ricci curvature. Despite the remarkable developments on this subject on a continuous space, much less is known when the underlying space is discrete, as in an undirected) graph. In recent work, Chow, Huang, Li and Zhou [6] considered Fokker-planck equations for a free energy function or a certain Markov process) defined on a finite graph. For a graph on N 2 vertices, they showed that the corresponding Fokker-Planck equation is a system of N nonlinear ordinary differential equations, defined on a Riemannian manifold of probability distributions. In fact, they point out that one could make different choices for inner products on the space of probability distributions resulting, in turn, in different Fokker-Planck equations for the same process. Furthermore, each of these systems of ordinary differential equations has a unique global equilibrium a Gibbs distribution and is a gradient flow for the free energy functional defined on a Riemannian manifold whose metric is closely related to certain classical Wasserstein metrics. We recall here, more formally, the approach of Chow et al [6]. Consider a graph G = V, E), where V = {a 1,, a N } is the set of vertices V 2, and E denotes the set of undirected) edges. For simplicity, assume that the graph is connected and is simple with no self-loops or multiple edges. Let Ni) := {j {1, 2,, N} {a i, a j } E} denote the neighborhood of a vertex a i V. Let Ψ = Ψ i ) N be a given potential function on V, where Ψ i is the potential on vertex a i. Further denote N M = {ρ = ρ i ) N R N ρ i = 1 and ρ i > for i = 1, 2,, N}, as the space of all positive probability distributions on V. Then the free energy functional has the following expression: for each ρ M, let N N 1.1) F ρ) := F Ψ,β ρ) = Ψ i ρ i + β ρ i log ρ i, where β > is the strength of white noise or the temperature. The free energy functional has a global minimizer, called a Gibbs density, and is given by 1.2) ρ i = 1 N K e Ψ i/β, where K = e Ψi/β. From a free energy viewpoint, Chow et al [6] endowed the space M with a Riemannian metric d Ψ, which depended on the potential Ψ as well as the structure of the graph. Then by considering the gradient flow of the free energy 1.1) on such a Riemannian manifold

3 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 3 M, d Ψ ), they obtained a Fokker-Planck equation on M: dρ i = Ψ j + β log ρ j ) Ψ i + β log ρ i ))ρ j j Ni),Ψ j >Ψ i 1.3) + Ψ j + β log ρ j ) Ψ i + β log ρ i ))ρ i j Ni),Ψ j <Ψ i + βρ j ρ i ) j Ni),Ψ j =Ψ i for i = 1, 2, Nsee Theorem 2 in [6]). From a stochastic process viewpoint, the work of Chow et al may be seen as a new interpretation of white noise perturbations to a Markov process on V. By considering the time evolution equation of its probability density function, they obtained another Fokker-Planck equation on M: dρ i = Ψ j + β log ρ j ) Ψ i + β log ρ i ))ρ j j Ni), Ψ j > Ψ i + Ψ j + β log ρ j ) Ψ i + β log ρ i ))ρ i j Ni), Ψ j < 1.4) Ψ i = Ψ j + β log ρ j ) Ψ i + β log ρ i )) + ρ j j Ni) j Ni) Ψ j + β log ρ j ) Ψ i + β log ρ i )) ρ i for i = 1, 2,, N, where Ψ i = Ψ i + β log ρ i for i = 1, 2,, N see Theorem 3 in [6]). For convenience, we call equations 1.3) and 1.4) Fokker-Planck equation I 1.3) and II 1.4) respectively. Both 1.3) and 1.4) share similar properties for β > see Theorem 2 and Theorem 3 in [6]): 1) Both equations are gradient flows of the same free energy on the same probability space M, but with different metrics. 2) The Gibbs distribution ρ = ρ i ) N, given by 1.2), is the unique stationary distribution of both equations in M. Furthermore, the free energy F attains its global minimum at the Gibbs distribution. 3) For both equations, given any initial condition ρ M, there exists a unique solution ρt) : [, ) M with the initial value ρ M, and ρt) satisfying the properties: a) the free energy F ρt)) decreases as time t increases, and b) ρt) ρ under the Euclidean metric of R N, as t +. There are differences between equations 1.3) and 1.4). Fokker-Planck equation I 1.3) is obtained from the gradient flow of the free energy F on the Riemannian metric space M, d Ψ ). However, its connection to a Markov process on the graph is not clear. On the other hand, Fokker-Planck equation II 1.4) is obtained from a Markov process subject to white noise perturbations. This equation can also be considered as a generalized gradient flow of the free energy on another metric space M, d Ψ) see Theorem 3 in [6]).

4 4 R. CHE, W. HUANG, Y. LI, AND P. TETALI However, the geometry of M, d Ψ) is not smooth. In fact, Chow et al showed that, in this case, M is divided into finite segments, and d Ψ is only smooth on each segment. By the above discussion, we know that the Gibbs distribution ρ = ρ i ) N is the unique global equilibrium of both equations 1.3) and 1.4) in M, and for any solution ρt) of both equations 1.3) and 1.4), ρt) will converge to global equilibrium ρ, under the Euclidean metric of R N, as t. A natural next question this raises is then that of the derivation of estimates, such as Oe ct ) for a suitable c >, on the rate of convergence to global equilibrium for solutions of both equations 1.3) and 1.4). Answering such a question is one of the main objectives of the present paper. The rates of convergence towards global equilibrium for the solution of these Fokker-Planck equations on a graph are investigated. We will prove that the convergence is indeed exponential. In [6], the authors introduced several metrics on the space of probability measures M, including d Ψ, d Ψ, an upper bound metric d M and a lower bound metric d m, where the latter two are independent of the choice of potential. These distances are obtained in a sense by discretizing Felix Otto s calculus there is a certain similarity between these distances and the 2-Wasserstein distance on the space of probability measures on R n. For example, the gradient flow of free energy functional defined using relative entropy) in these metric spaces gives rise to the discrete Fokker-Planck equation in [6]. It is worth mentioning that the geodesic of d Ψ is a discretization of the geodesic equation in 2-Wasserstein distance on the space of probability measures on R n. For these reasons, sometimes we refer to these as discrete 2-Wasserstein distances. As an important application of our convergence result, Talagrand-type inequalities are proved. We will show that the 2-Wasserstein distance is bounded from above by the relative entropy: that for all ν absolutely continuous with respect to µ, it holds: d 2 mν, µ) KHν µ), where K only depends only on the reference measure) µ. In recent years, there has been considerable interest in deriving such inequalities in various spaces, with the purpose of studying geometric inequalities connected to concentration of measure and other phenomenon. On a Riemannian manifold, Otto and Villani [18] showed inter alia) that a logarithmic Sobolev inequality implied a Talagrand inequality; this work was soon generalized and simplified by Bobkov, Gentil and Ledoux [4]; the latter provided simpler proofs of several previously known results concerning log- Sobolev and transport inequalities. See also [3] for an earlier work which along with [18]) inspired much of the research in this topic. In subsequent work, Lott and Villani [12] used the Hamilton-Jacobi semigroup approach of Bobkov et al [4] in showing that a Talagrand inequality on a measured length space implied a global Poincaré inequality, as well as in obtaining conversely), that spaces satisfying a certain doubling condition, a local Poincaré inequality and a log-sobolev inequality satisfied a Talagrand inequality. In discrete spaces, such inequalities are less understood. In part, the lack of a suitable 2-Wasserstein W 2 ) distance between probability measures on a graph has slowed this progress. M. Sammer and the last author of this work observed see [21, 22] for a proof) that a derivation of Otto-Villani goes through in the context of a finite graph in yielding the implication that a weaker) modified log-sobolev inequality implies a weaker) Talagrand-type inequality, relating a W 1 -distance rather than a W 2 ) and the relative entropy.

5 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 5 In the following, we obtain in fact two versions of a Talagrand-type inequality. The first one is only locally true, which means that the parameter K also depends on the range of ν it is true for all measures ν in a compact neighborhood of µ, but may not be true if ν is arbitrarily far away from µ. The global Talagrand inequality holds for the lower bound metric. If the graph G is simple and connected, then there exists a parameter K that only depends on µ and certain parameters of G. Establishing such an inequality for a suitable notion of a W 2 - distance on a discrete space continues to be intriguing; particularly, since various people have independently observed that a literal translation of such an inequality, borrowed from the continuous case, need not hold even on a 2-point discrete space see e.g., [15,?]). However, our results suggest that the metrics introduced in [6] have a further similarity with the W 2 -distance on the space of probability measures of the length space. It is known that on R n, Talagrand inequality is implied by the log-sobolev inequality. It remains to be seen, however, whether such an implication is true in the present case. Independently, a related class of metrics has been studied by Mielke in [16] and Maas in [15], which are similar to the Riemannian metrics in [6] with a constant potential. In the setting of both [16] and [15], the graphs are assumed to be associated with an irreducible and reversible Markov kernel. After essentially finishing this paper, the authors have been informed that functional inequalities including modified Talagrand inequality and modified logarithmic Sobolev inequalities associated with the Riemannian metric studied in [15, 16] are independently investigated in [7]. 2. Preliminaries In this section, we recall some definitions in graph theory. A graph is an ordered pair G = V, E) where V = {a 1,, a N } is the set of vertices and E is the set of edges. We further assume that the graph G is a simple graph that is, there are no self loops or multiple edges) with V 2, and G is connected. A weighted graph G, w) is a pair consisting of a graph G = V, E) and a positive real-valued function w of its edges. The function w is most conveniently described as an V -by- V, symmetric, nonnegative matrix w = w ij ) with the property that w ij > if and only if a i, a j ) E. Given a graph G = V, E) with V = {a 1, a 2,, a N }, we consider all positive probability distributions on V : { } N M = ρ = ρ i ) N R N ρ i = 1 and ρ i > for i {1, 2,, N}. For µ = µ i ) N M and any map f : V R, recall the L 2 µ)-norm of f with respect to µ, denoted by f 2,µ, and given by: N f 2 2,µ := fa i )) 2 µ i Let ν = ν i ) N M, then the relative entropy Hν µ) of ν with respect to µ is defined by: N Hν µ) = ν i log ν i, µ i

6 6 R. CHE, W. HUANG, Y. LI, AND P. TETALI and we measure the distance between the density of) ν and µ using: ν N µ 1 2 2,µ := ν i 1) 2 µ i. µ i Given a graph G = V, E), its Laplacian matrix is defined as: LG) := D A, where D is a diagonal matrix with d ii = dega i ) number of edges at a i ), and A is the adjacency matrix A ij = 1 if and only if {a i, a j } E). As G is a connected simple graph, it is well known that LG) has one eigenvalue and V 1 positive eigenvalues. Given a weighted graph G, w), its weighted Laplacian matrix is defined as LG, w) = diagδ 1, δ 2,, δ V ) w with δ i denoting the ith row sum of w. It is well known that LG, w) also has one eigenvalue and V 1 positive eigenvalues. The second smallest eigenvalue λ 2 of LG) resp. LG, w)) is called the spectral gap of G resp.g, w)). We remind readers that there are various standard ways to bound the spectral gap of a graph. For example for the spectral gap λ 2 of LG), see [2] for the bound λ 2 d max d 2 max d 2 min, where d max and d min are the maximum and minimum degrees of vertices in G; similarly see [13], for 2N λ NN 1)d 2Md, where N is the number of vertices, M is the number of edges, and d is the diameter of G; or [27] for the bound, λ 2 21 cos π N )). 3. The Trend towards Equilibrium The rate of convergence towards global equilibrium for the solution of Fokker-Planck equations 1.3) and 1.4) in weighted L 2 norm are estimated in this section. We will prove that such convergence is exponential. In addition, the relative entropy with respect to the global equilibrium ) also has exponential decay Convergence in weighted L 2 norm. The following is our first main result. Theorem 3.1. Let G = V, E) be a graph with its vertex set V = {a 1, a 2,, a N }, edge set E, a given potential Ψ = Ψ i ) N on V and a constant β >. If ρt) = ρ i t)) N : [, ) M is the solution of the Fokker-Planck equation I 1.3), with the initial value ρ o = ρ o i ) N M, then there exists a constant C = Cρ o ; G, Ψ, β) > such that 3.1) ρt) ρ 1 2 2,ρ ρo ρ 1 2 2,ρ e Ct, where ρ = ρ i ) N is the Gibbs distribution given by 1.2). In particular, ρt) exponentially converges to global equilibrium: the Gibbs distribution ρ under the Euclidean metric of R N as t.

7 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 7 Proof. Given initial value ρ o = ρ o i ) N M. Let ρt) = ρ i t)) N : [, ) M be the solution of Fokker-Planck equation I 1.3) with initial value ρ o M. For t, we define Lt) = ρt) ρ 1 2 2,ρ = N ρ i t) ρ i ) 2, where ρ = ρ i ) N is the Gibbs distribution given by 1.2). Now for t > by 1.3) we have dlt) N 2ρ i t) ρ i ) dρ i t) = ρ i = + N 2ρ i t) ρ i ) ρ i j Ni),Ψ j <Ψ i j Ni),Ψ j >Ψ i ρ i Ψj + β log ρ j t)) Ψ i + β log ρ i t)) ) ρ j t) Ψj + β log ρ j t)) Ψ i + β log ρ i t)) ) ρ i t) ) + βρ j t) ρ i t)). j Ni),Ψ j =Ψ i Note that Ψ j Ψ i = β log ρ j +β log ρ i for i, j {1, 2,, N} and ρ j = ρ i when Ψ j = Ψ i. Combining this with the above equality, we have dlt) N 2ρ i t) ρ i ) = β log ρ j + β log ρ j t)) β log ρ i + β log ρ i t)) ) ρ j t) + ρ i j Ni),Ψ j <Ψ i + β ρ jt) ρ j Ni),Ψ j =Ψ j i N = + j Ni),Ψ j >Ψ i β log ρ j + β log ρ j t)) β log ρ i + β log ρ i t)) ) ρ i t) 2ρ i t) ρ i ) ρ i βlog ρ jt) ρ j Ni),Ψ j <Ψ j i ρ it) ) ρ i + ρ ) j ρ i 2 βlog ρ jt) ρ j Ni),Ψ j >Ψ j i log ρ it) )ρ ρ i t) + i log ρ it) )ρ ρ j t) i β ρ jt) ρ j Ni),Ψ j =Ψ j i ρ it) ) ρ i + ρ ) j ρ i 2

8 8 R. CHE, W. HUANG, Y. LI, AND P. TETALI We denote η i t) as ρ it) ρ i ρ i N dlt) = + for t. Then the above equation can be written as 2η i t) j Ni),Ψ j <Ψ i β j Ni),Ψ j >Ψ i β log1 + ηj t)) log1 + η i t)) ) ρ j t) log1 + ηj t)) log1 + η i t)) ) ρ i t) + βη j t) η i t)) ρ i + ρ j 2 j Ni),Ψ j =Ψ i For edge {a i, a j } E with Ψ j > Ψ i, 2η i βlog1 + η j ) log1 + η i ))ρ j will be in the above sum at vertex a i ; 2η j βlog1 + η i ) log1 + η j ))ρ j will be in the above sum at vertex a j. So we can write the above equality as dlt) = 2βlog1 + η j t)) log1 + η i t)))η j t) η i t))ρ j t) {a i,a j } E,Ψ j >Ψ i 2βη j t) η i t)) 2 ρ i + ρ j 3.2). 2 {a i,a j } E,Ψ j =Ψ i Using 3.2) and the following inequality min{ 1 a, 1 log a log b } max{ 1 b a b a, 1 b } for a >, b > with a b, we have dlt) 2βη j t) η i t)) 2 1 min{ 1 + η i t), η j t) }ρ jt) {a i,a j } E,Ψ j >Ψ i 2βη j t) η i t)) 2 ρ i + ρ j 2 {a i,a j } E,Ψ j =Ψ i = 2βη j t) η i t)) 2 min{ ρ i ρ i t), ρ j ρ j t) }ρ jt) {a i,a j } E,Ψ j >Ψ i 2βη j t) η i t)) 2 ρ i + ρ j 3.3). 2 {a i,a j } E,Ψ j =Ψ i For b = b i ) N R N, we let mb) = min{b i : 1 i N} and Mb) = max{b i : 1 i N}. Put At) = 2β mρt)) Mρt)) mρ ) for t. Then At) > and by 3.3) we have dlt) 3.4) At) η j t) η i t)) 2 ). {a i,a j } E Next we use the following claims whose proofs appear after the proof of the present theorem), relating the above right hand side to the spectral gap of the Laplacian matrix LG) of graph G. ).

9 Claim 3.2. CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 9 {a i,a j } E η j t) η i t)) 2 λ 2 2Mρ ) Lt), where Mρ ) is the maximal entry of ρ which is less than 1, and λ 2 is the second smallest eigenvalue of the Laplacian matrix LG) of G, or the spectral gap of G. We need the following definition before stating the next claim. Let us denote M = max{e 2 Ψ i : i = 1, 2,, N}, ɛ = 1, and { } ɛ 1 = 1 2 min ɛ, min{ρ 1 + 2M) 1 i : i = 1,, N}. β ) For l = 2, 3,, N 1, we let ɛ l 1 ɛ l = M) 1 β We define l B = {q = q i ) N M : q ir 1 ɛ l where l {1,, N 1}, r=1 1 i 1 < < i l N}. Then B is a compact subset of M with respect to the Euclidean metric, with l intb) = {q = q i ) N M : q ir < 1 ɛ l, where l {1,, N 1}, and ρ intb). We have r=1 1 i 1 < < i l N}. Claim 3.3. ρt) B for all t. Using Claim 3.2 and 3.4), we have 3.5) We define dlt) 3.6) C = βλ 2 mρ ) Mρ ) λ 2 At) Lt). 2Mρ ) 1 ɛ L 1 ɛ 1. Clearly C > is dependent on ρ o as well as on G, Ψ, β, that is C = Cρ o ; G, Ψ, β). By the definition of B and Claim 3.3, we have At) = 2β mρt)) Mρt)) mρ ) 2βmρ ) min{ mq) Mq) : q B} 2βmρ ) 1 ɛ L 1 ɛ 1 = 2Mρ ) C λ 2

10 1 R. CHE, W. HUANG, Y. LI, AND P. TETALI for t. Combining this with 3.5), we get dlt) CLt) for t >. This implies that Lt) L)e Ct for t. Since L) = ρ) 1 2 ρ 2,ρ, we have 3.1), completing the proof of the theorem, modulo the claim see below). Remark 3.4. Given a graph G = V, E), a potential Ψ = Ψ i ) N on V and a constant β >, the positive constant C = Cρ o ; G, Ψ, β) given by 3.6) is dependent on the initial value ρ o M. In fact Cρ o ; G, Ψ, β), when the initial distribution ρ o converges to the boundary of M. We now write a simple observation yielding the first claim used in the proof of the above theorem. Proof of Claim 3.2. Indeed we have {a i,a j } E 1 2Mρ ) η j t) η i t)) 2 = 1 2 N j Ni) N j Ni) η j t) η i t)) 2 η j t) η i t)) 2 ρ i λ 2 2Mρ ) V ar ρ ηt)) the last inequality comes from the Poincare-type inequality See for example [5]). which N N V ar ρ ηt)) = ρ i ηi 2 t) ρ i η i t)) 2 Note that and Hence we have N ρ i η i t) = {a i,a j } E N ρ i t) N ρ i = N ρ i ηi 2 t) = Lt) η j t) η i t)) 2 λ 2 2Mρ ) Lt). Finally, we present the proof of the second claim. Proof of Claim 3.3. We follow closely the argument in the proof of Theorem 4.1 in [6]. Since ρ intb), it is sufficient to show for any q B, the solution qt) through q remains in intb) for small t >. Let q = q i ) N B and qt) : [, cq)) M be the solution to the equation 1.3) with initial value q on its maximal interval of existence. In order to show qt) intb) for small t >, it is sufficient to show that for any l {1, 2,, N 1} and 1 i 1 < i 2 < i l N, one has l q ir t) < 1 ɛ l, r=1 In

11 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 11 for sufficiently small t >. Given l {1, 2,, N 1} and 1 i 1 < i 2 < i l N, since q B, we have l q ir 1 ɛ l. r=1 Then there are two cases. The first one is l q ir < 1 ɛ l. It is clear that for small enough t > by continuity. The second case is r=1 l q ir t) < 1 ɛ l, r=1 l q ir = 1 ɛ l. Let A = {i 1, i 2,, i l } and A c = {1, 2,, N} \ A. Then for any j A c, l 3.7) q j 1 q ir ) = ɛ l. Since q B, we have r=1 r=1 l 1 q sj 1 ɛ l 1, for any 1 s 1 < s 2 < < s l 1 N. Hence for each i A, 3.8) q i 1 ɛ l 1 ɛ l 1 ) = ɛ l 1 ɛ l. j=1 Combining equations 3.7),3.8) and the fact one has, for any i A, j A c, ɛ l ɛ l M) 1 β 3.9) Ψ j Ψ i + βlog q j log q i ) Ψ j Ψ i + βlog ɛ l logɛ l 1 ɛ l )) log 2. For {a i, a j } E, we set q j if Ψ i < Ψ j 3.1) C{a i, a j }) = q i if Ψ i > Ψ j. q i q j log q i log q j if Ψ i = Ψ j Clearly, C{a i, a j }) > for {a i, a j } E. Since the graph G is connected, there exists i A, j A c such that {a i, a j } E. Thus 3.11) C{a i, a j }) C{a i, a j }) >. i A,j A c,{a i,a j } E,

12 12 R. CHE, W. HUANG, Y. LI, AND P. TETALI Now by 3.9) and 3.11), one has d l q ir t) t= = C{a i, a j }) Ψ j Ψ i + βlog q j log q i ) )) r=1 i A j Ni) = C{a i, a j }) Ψ j Ψ i + βlog q j log q i ) ) + i A j A Ni) C{a i, a j }) Ψ j Ψ i + βlog q j log q i ) )) j A c Ni) = i A i A Combining this with the fact it is clear that j A c Ni) j A c Ni) = log 2 C{a i, a j }) Ψ j Ψ i + βlog q j log q i ) )) C{a i, a j }) log 2 ) i A,j A c,{a i,a j } E C{a i, a j }) log 2 <. l q ir = 1 ɛ l, r=1 l q ir t) < 1 ɛ l, r=1 C{a i, a j }) ) for sufficiently small t >. This finishes the proof of Claim 3.3. Using the same technique, we have the following second main result. Theorem 3.5. Let G = V, E) be a graph with its vertex set V = {a 1, a 2,, a N }, edge set E, a given potential Ψ = Ψ i ) N on V and a constant β >. If ρt) : [, ) M is the solution of Fokker-Planck equation II 1.4), with the initial value ρ o = ρ o i ) N M, then 3.12) ρt) ρ 1 2 2,ρ ρo ρ 1 2 2,ρ e Ct, where ρ = ρ i ) N is the Gibbs distribution given by 1.2) and C = βλ 2 min{ρ i :1 i N} max{ρ i :1 i N}, where λ 2 is the spectral gap of G. In particular, ρt) exponentially converges to global equilibrium: the Gibbs distribution ρ under the Euclidean metric of R N, as t. Proof. Given initial value ρ = ρ i ) N M. Let ρt) = ρ i t)) N : [, ) M be the solution of Fokker-Planck equation II 1.4) with initial value ρ M. For t, we define Lt) = ρt) 1 2 ρ 2,ρ = N ρ i t) ρ i ) 2, ρ i

13 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 13 where ρ = ρ i ) N is the Gibbs distribution given by 1.2). Now for t >, by 1.4), we have dlt) N 2ρ i t) ρ i ) dρ i t) = ρ i = + = + N 2ρ i t) ρ i ) Ψj + β log ρ ρ j t)) Ψ i + β log ρ i t)) ) ρ j t) i j Ni), Ψ j > Ψ i Ψj + β log ρ j t)) Ψ i + β log ρ i t)) ) ρ i t)) j Ni), Ψ j < Ψ i N 2ρ i t) ρ i ) βlog ρ jt) ρ i ρ j Ni), Ψ j > Ψ j i βlog ρ jt) ρ j Ni), Ψ j < Ψ j i log ρ ) it) )ρ ρ i, i log ρ it) )ρ ρ j i the last equality comes from the fact Ψ j Ψ i = β log ρ j +β log ρ i for i, j {1, 2,, N}. Denoting ρ it) ρ i ρ i N dlt) = + = by η i t), for t >, the equation will be written as 2βη i t) j Ni), Ψ j < Ψ i j Ni), Ψ j > Ψ i {a i,a j } E, Ψ j > Ψ j 2β log1 + ηj t)) log1 + η i t)) ) ρ j t) log1 + ηj t)) log1 + η i t)) ) ρ i t)) log1 + ηj t)) log1 + η i t)) ) η j t) η i t) ) ρ j t). Moreover note that η i t) = η j t) when Ψ i = Ψ j, we have dlt) = log1 + ηj t)) log1 + η i t)) ) η j t) η i t) ) ρ j t) 3.13) {a i,a j } E, Ψ j > Ψ i 2β {a i,a j } E, Ψ j = Ψ i 2β Using 3.13) and the following inequality ηj t) η i t) ) 2 ρ i + ρ j 2 min{ 1 a, 1 log a log b } b a b. max{ 1 a, 1 b },

14 14 R. CHE, W. HUANG, Y. LI, AND P. TETALI for a >, b > with a b, we have dlt) ηj t) η i t) ) 2 1 min{ 1 + η i t), η j t) }ρ jt) 3.14) = {a i,a j } E, Ψ j > Ψ i 2β {a i,a j } E, Ψ j = Ψ i 2β {a i,a j } E, Ψ j > Ψ i 2β {a i,a j } E, Ψ j = Ψ i 2β ηj t) η i t) ) 2 ρ i + ρ j 2 ηj t) η i t) ) 2 ρ min{ i ρ i t), ρ j ρ j t) }ρ jt) ηj t) η i t) ) 2 ρ i + ρ j 2 Now note that ρ i ρ j when Ψ ρ i t) ρ j t) j > Ψ i, hence min{ ρ i, ρ j }ρ ρ i t) ρ j t) jt) = ρ j when Ψ j > Ψ i. Combining this with 3.14), we have dlt) ηj t) η i t) ) 2 ρ j 3.15) {a i,a j } E, Ψ j > Ψ i 2β 2β ηj t) η i t) ) 2 ρ i + ρ j 2 {a i,a j } E, Ψ j = Ψ i 2β min{ρ i : 1 i N} η j t) η i t)) 2.. {a i,a j } E By the same argument as in Claim 3.2 used in the proof of Theorem 3.1), we have η j t) η i t)) 2 λ 2 2 max{ρ i : 1 i N}Lt). {a i,a j } E Combining this with 3.15), we get dlt) CLt) for t >, where C = βλ 2 min{ρ i :1 i N} max{ρ i :1 i N}. This implies Lt) L)e Ct for t, that is 3.12) is true, completing the proof of the theorem Exponential decay of the relative entropy. In the following, we show that the entropy decay rate of the Fokker-Planck Equation II) on G = V, E) with its vertex set V = {a 1, a 2,, a N }, edge set E, a given potential Ψ = Ψ i ) N on V and a constant β > can be bounded in terms of the modified logarithmic Sobolev constant also known as the entropy constant ) γ := γ G) of the underlying graph G: the optimal γ > such that 3.16) 2γ Entf) Ef, log f), over all f : V R, with f > ; recall here the standard notation for the Entropy functional and the Dirichlet form: Entf := Ent ρ f := E ρ f log f) E ρ f) loge ρ f).

15 and CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 15 Ef, log f) = N j Ni) log fai ) log fa j ) ) fa i ) fa j ) ) ρ i. where ρ = ρ i ) N is the Gibbs distribution given by 1.2). See [5] where this constant was denoted as ρ ) and references therein, for more information on γ of a graph and that of a Markov kernel on G. Theorem 3.6. Let G = V, E) be a graph, with its vertex set V = {a 1, a 2,, a N }, edge set E, a given potential Ψ = Ψ i ) N on V and a constant β >. If ρt) = ρ i t)) N : [, ) M is the solution of Fokker-Planck equation II 1.4) with the initial value ρ o = ρ o i ) N M, then Hρt) ρ ) Hρ ρ )e ct for t, where ρ = ρ i ) N is the Gibbs distribution given by 1.2) and c = βγ min{ρ i :1 i N} max{ρ i :1 i N}. Proof. Given ρ o = ρ o i ) N M. Let ρt) = ρ i t)) N : [, ) M be the solution of Fokker-Planck equation II 1.4) with the initial value ρ. Recall that the relative entropy of ρ = ρ i ) N M with respect to ρ : Hρ ρ ) = N ρ i log ρ i ρ i Since the Gibbs distribution is given by ρ i = 1 K e Ψ i/β, we also have Ψ i = β log ρ i β log K. For t, let ft) = ρt)/ρ, i.e, ft)a i ) = ρ i t)/ρ i for i = 1, 2,, N, we rewrite the relative entropy as usual: Entft) := Ent ρ ft) = E ρ f log f) E ρ f) loge ρ f) = Hρt) ρ ). We write simply f i t) = ft)a i ). Then the Fokker-Planck equation II 1.4) becomes dρ i t) = β log f j t) log f i t))ρ j t) + ) log f j t) log f i t))ρ i t). j Ni), f j t)>f i t). j Ni), f j t)<f i t)

16 16 R. CHE, W. HUANG, Y. LI, AND P. TETALI Observing that a b log a log b when a > b >, we bound the entropy decay by a proceeding as follows. d N ) ρ i t) log ρ it) dentft)) ρ i = N dρ i t) N dρ i t) N dρ i t) = log f i t) + = log f i t) = β = β β = β N log f i {a i,a j } E f j t)<f i t) {a i,a j } E f j t)<f i t) {a i,a j } E f j t)<f i t) j Ni), f j t)>f i t) log f j t) log f i t))ρ j t) + log f i t) log f j t)) 2 ρ i t) 1 min{ρ i : 1 i N} 2 max{ρ i : 1 i N}β = 1 min{ρ i : 1 i N} 2 max{ρ i : 1 i N}β log fi t) log f j t) ) f i t) f j t)) ρ i t) f i t) log fi t) log f j t) ) f i t) f j t))ρ i {a i,a j } E f j t)<f i t) {a i,a j } E j Ni), f j t)<f i t) ) log f j t) log f i t))ρ i t) log fi t) log f j t) ) f i t) f j t))ρ i + ρ j) log fi t) log f j t) ) f i t) f j t))ρ i + ρ j) = 1 min{ρ i : 1 i N} log f), 2 max{ρ i : 1 i N}βEf, where E, ) is the Dirichlet form of G = V, E) with respect to the measure ρ on V. Now using the definition of the modified log-sobolev constant 3.16), we conclude that resulting in : dentf) centf), Entft)) Entf))e ct, that is, Hρt) ρ ) Hρ ρ )e ct min{ρ for t, where c = βγ i :1 i N} max{ρ i :1 i N}. This completes the proof of Theorem. 4. Talagrand-type Inequalities 4.1. Discrete Wasserstein-type metric on M. Consider a graph G = V, E) with V = {a 1, a 2,, a N }. As the collection of positive probability distributions on V, space M is defined as in the beginning of Section 2.

17 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 17 The tangent space T ρ M at ρ M has the form { T ρ M = σ = σ i ) N R N } N σ i =. It is clear that the standard Euclidean metric on R N, d, is also a Riemannian metric on M. Let Φ : M, d) R N, d) be an arbitrary smooth map given by: Φρ) = Φ i ρ)) N, ρ M. In the following, we will endow M with a metric d Φ, which depends on Φ and the structure of G. We consider the function r 1 r 2 log r 1 log r 2 and extend it to the closure of the first quadrant in the plane. Denote r 1 r 2 log r 1 log r 2 if r 1 r 2 and r 1 r 2 > er 1, r 2 ) = if r 1 r 2 =. r 1 if r 1 = r 2 It is easy to check that er 1, r 2 ) is a continuous function on the first quadrant and satisfies min{r 1, r 2 } er 1, r 2 ) max{r 1, r 2 }. For simplicity, we use its original form instead of the function er 1, r 2 ) in the present paper. The equivalence relation on R N is defined as p q if and only if p 1 q 1 = p 2 q 2 = = p N q N, and let W be the quotient space R N /. In other words, for p R N we consider its equivalent class [p] = {p 1 + c, p 2 + c,, p N + c) : c R}, and all such equivalent classes form the vector space W. For a given Φ, and [p] = [p i ) N ] W, we define an identification τ Φ [p]) = σ from W to T ρ M by, 4.1) σ = plg, wρ)), where wρ) = {w ij ρ)} N i,j=1 is the weight associated to the original graph G: ρ i if Φ i ρ) > Φ j ρ), {a i, a j } E ρ j if Φ i ρ) < Φ j ρ), {a i, a j } E w ij ρ) = ρ i ρ j log ρ i log ρ j if Φ i ρ) = Φ j ρ), {a i, a j } E otherwise and LG, wρ)) is the weighted Laplacian matrix of weighted graph G, wρ)). It is not hard to check that τ Φ [p]) is a linear isomorphism between T ρ M and W see Lemma 2 in [6]). Hence σ T ρ M can be rewritten as equivalent classes on R N. For simplicity, we say σ [p] = [p i ) N ] if [p] := τ 1 Φ σ) W.

18 18 R. CHE, W. HUANG, Y. LI, AND P. TETALI We note that this identification 4.1) depends on Φ, the probability distribution ρ and the structure of the graph G. Definition 4.1. By the above identification 4.1), we define an inner product on T ρ M by: N N gρ Φ σ 1, σ 2 ) = p 1 i σi 2 = p 2 i σi 1. It is easy to check that this definition is equivalent to g Φ ρ σ 1, σ 2 ) = p 1 LG, wρ))p 2 ) T, for σ 1 = σ 1 i ) N, σ 2 = σ 2 i ) N T ρ M, and [p 1 ], [p 2 ] W satisfying In particular, σ 1 [p 1 ] and σ 2 [p 2 ]. 4.2) g Φ ρ σ, σ) = plg, wρ))p T for σ T ρ M, where σ [p]. The associated distance d Φ, ) on M is given by d Φ ρ 1, ρ 2 ) = inf γ Lγt)) where γ : [, 1] M ranges over all continuously differentiable curve with γ) = ρ 1, γ1) = ρ 2. The arc length of γ is given by 1 Lγt)) = gγt) Φ γt), γt)). This gives the metric space M, d Φ ). Next, we show that the metric d Φ is lower bounded. Given ρ = ρ i ) N M, We consider a new identifications 4.3) σ = plg, w m ρ)), where w m ρ) = {wij m ρ)} N i,j=1 is the weight associated to the original graph G: { wij m max{ρ i, ρ j } if {a i, a j } E ρ) =. otherwise and LG, w m ρ)) is the weighted Laplacian matrix of weighted graph G, w m ρ)). Similar to the identification 4.1), identifications 4.3) is linear isomorphisms between T ρ M and W. Furthermore, they induce inner product g m ρ, ) on T ρ M. It is not hard to see that the map ρ g m ρ is smooth. By using the inner products g m ρ, we can obtain distance d m, ) on M. Then M, d m ) is smooth Riemannian manifold. It is shown in [6, Lemma 3.4] that for any smooth map Φ : M, d) R N, d) and ρ 1, ρ 2 M, 4.4) d m ρ 1, ρ 2 ) d Φ ρ 1, ρ 2 ). Now we consider two choices of the function Φ which are related to Fokker-Planck equation I 1.3) and II 1.4) respectively. Let the potential Ψ = Ψ i ) N on V be given and β >, where Ψ i is the potential on vertex a i.

19 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 19 It then follows from [6, Section 4] that Fokker-Planck equation I 1.3) is the gradient flow of free energy F ρ) on the Riemannian manifold M, d Ψ ), i.e. let Φρ) Ψ where ρ M. Fokker-Planck equation II 1.4) is related to inner product g Ψ ρ, where the new potential Ψρ) = Ψ i ρ)) N is defined by Ψ i ρ) = Ψ i + β log ρ i. Since g Ψ ρ is a piecewise smooth function with respect to ρ, the space M, d Ψ) is a union of finitely many smooth Riemannian manifolds. Fokker-Planck equation II 1.4) can also be seen as the generalized gradient flow of F ρ) on the metric space M, d Ψ) see [6, Section 5]). By 4.4), d Ψ and d Ψ are lower bounded by d m Talagrand-type inequalities. We are now ready to prove the Talagrand-type inequalities. Theorem 4.2. Let G = V, E) be a graph with its vertex set V = {a 1, a 2,, a N } and edge set E. For each µ = µ i ) N M and a compact subset B of M with respect to the Euclidean metric, there exist a potential function Ψ = Ψ i ) N on V and a constant K = KB, µ, G) > such that for any ν = ν i ) N B, we have the following Talagranype inequality d 2 Ψµ, ν) KHν µ), where Hν µ) = N ν i log ν i µ i. Proof. Given µ = µ i ) N M and a compact subset B of M with respect to the Euclidean metric. Let Ψ i = log µ i for i = 1, 2,, N and β = 1. Then F ν) = Hν µ) for any ν M. In the following, we are going to show that there is a constant K = KB, µ, G) > such that d 2 Ψν, µ) KHν µ) = KF ν) for any ν B. Firstly using 4.1) and 4.2), for σ T ρ M we have and σ 2 = plg, wρ))lg, wρ)) T p T g Φ ρ σ, σ) = plg, wρ))p T, where σ [p] and is the standard Euclidean norm on R N. Since LG, wρ)) is a real symmetric matrix, we decompose LG, wρ)) into LG, wρ)) = QΛQ T where Λ is a diagonal matrix whose diagonal entries are eigenvalues of LG, wρ)), and Q is a real orthogonal matrix. Let w = pq, then we have 4.5) g Ψ ρ σ, σ) = plg, wρ))p T = wλw T and σ 2 = plg, wρ))lg, wρ)) T p T = wλ 2 w T.

20 2 R. CHE, W. HUANG, Y. LI, AND P. TETALI Denote λ 2 ρ) and λ N ρ) the second smallest eigenvalue and largest eigenvalue of LG, wρ)) respectively. Since LG, wρ)) has one eigenvalue and N 1 positive eigenvalues, it is not hard to see 1 λ N ρ) σ 2 gρ Ψ σ, σ) 1 λ 2 ρ) σ 2 by 4.5). Let us denote M = max{e 2 Ψ i : i = 1, 2,, N}, ɛ = 1, and { } ɛ 1 = 1 2 min ɛ, min min{ρ 1 + 2M) 1 i : i = 1,, N}, β ) ρ i ) N B where ɛ 1 > as B is compact. For l = 2, 3,, N 1, we let We define D = {q = q i ) N M : ɛ l = ɛ l M) 1 β l q ir 1 ɛ l where l {1,, N 1}, r=1 1 i 1 < < i l N}. Then D is a compact subset of M with respect to the Euclidean metric and with l intd) = {q = q i ) N M : q ir < 1 ɛ l, where l {1,, N 1}, and B intd). Let r=1 1 i 1 < < i l N}. C 1 = max { 1 ρ D λ 2 ρ) } and C 2 = min { 1 ρ D λ N ρ) }, Since λ 2, λ N : M, + ) are continuous and D is compact with respect to the Euclidean metric on M, we have < C 2 C 1 < +. It is clear that C 1, C 2 depend only on B, µ and G, and 4.6) C 2 σ 2 g Ψ ρ σ, σ) C 1 σ 2 for any ρ D, σ T ρ M.

21 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 21 Now for ν B, let ρt) = ρ i t)) N : [, + ) M is the solution of the Fokker-Planck Equation 1.3) for β = 1: dρ i = Ψ j + log ρ j ) Ψ i + log ρ i ))ρ j j Ni),Ψ j >Ψ i + Ψ j + log ρ j ) Ψ i + log ρ i ))ρ i j Ni),Ψ j <Ψ i + ρ j ρ i ) j Ni),Ψ j =Ψ i with initial value ν, that is, ρ) = ν. Since ν intd), we have ρt) D for all t, which s proof is similar to the proof of Claim 3.3. Since the Gibbs distribution given by 1.2) is µ, Theorem 3.1 and 3.6) imply that there exists a constant C = Cµ, B, G) > such that N ρ i t) µ i ) 2 µ i N ν i µ i ) 2 µ i )e Ct for all t, Moreover let m = min{µ i : 1 i N} and M = max{µ i : 1 i N}, then ρt) µ 2 M m µ ν 2 e Ct for all t. Set T = 1 C log 4M m ). One obtains ρt ) µ µ ν µ ρt ) 2 + ρt ) ν 2 ), which implies ρt ) ν 2 ρt ) µ 2. So after time T, the solution of equation 1.3) traveled at least half of the Euclidean distance from ν to µ. Moreover since the Fokker-Planck equation equation 1.3) is the gradient flow of free energy F under the metric d Ψ, ) see Equation 31) and Theorem 2 in [6]), we have df ρt)) = g Ψ ρt) ρt), ρt)) for t >. By Integrating the previous equality from to T, we have T F ν) F ρt )) = gρt) Ψ ρt), ρt)) 1 T T gρt) Ψ ρt), ρt)))2 1 T T C2 ρt) ) 2 C 2 T ν ρt ) 2.

22 22 R. CHE, W. HUANG, Y. LI, AND P. TETALI the last second inequality comes from 4.6) and the fact that ρt) D. At the same time, F ν) F ρt )) = T 1 T T g Ψ ρt) ρt), ρt)) 1 T d2 Ψν, ρt )). gρt) Ψ ρt), ρt)))2 Let st) = tρt ) + 1 t)µ for t [, 1]. Since D is a convex subset of R N and ρt ), µ D. we have st) D for t [, 1]. Thus 1 d 2 ΨρT ), µ) gst) Ψ ρt ) µ, ρt ) µ))2 1 C1 ρt ) µ 2 ) 2 = C 1 ρt ) µ 2 the last second inequality comes from 4.6) and the fact that st) D. This gives us the bounds and In conclusion, d 2 ΨρT ), µ) C 1 ρt ) µ 2 C 1 ρt ) ν 2 T C 1 C 2 F ν) F ρt ))) T C 1 C 2 F ν), d 2 Ψν, ρt )) T F ν) F ρt ))) T F ν). d 2 Ψµ, ν) 2d 2 Ψµ, ρt )) + 2d 2 ΨρT ), ν) 2T C 1 C 2 + 2T )F ν) = KHν µ), where K = 2T C 1 C 2 + 2T ) is a parameter which only depends on B, G and µ. The other Talagrand-type inequality is for the lower bound metric d m, ). Theorem 4.3. Let G = V, E) be a graph with its vertex set V = {a 1, a 2,, a N } and edge set E. Let D be the maximal degree of G and λ 2 be the spectral gap of G. Given µ = µ i ) N M. Let m = min{µ i : 1 i N} and M = max{µ i : 1 i N}. Then for any ν = ν i ) N M, we have the following Talagrand-type inequality where K = MDN 3 +4) 2λ 2 m 4.1. d 2 mν, µ) KHν µ) log 18M ) and d m 3 m, ) is the lower bounded metric defined in Section Proof. Let β = 1, Ψ i = log µ i and Ψ i ρ) = log µ i + log ρ i for i = 1, 2,, N. Then F ρ) = Hρ µ) for ρ M. In the following, we are going to show that there is a constant K = Kµ, G) > such that for any ν B. d 2 mν, µ) KHν µ) = KF ν)

23 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 23 Now for ν B, let ρt) = ρ i t)) N : [, + ) M is the solution of the Fokker-Planck equation II) 1.4) with β = 1, dρ i = Ψ j + log ρ j ) Ψ i + log ρ i ))ρ j j Ni), Ψ j > Ψ i + Ψ j + log ρ j ) Ψ i + log ρ i ))ρ i. j Ni), Ψ j < Ψ i with initial value ν, that is, ρ) = ν. Let m = min{µ i : 1 i N}, M = max{µ i : 1 i N}. Since the Gibbs distribution given by 1.2) is µ, Theorem 3.5 implies N ρ i t) µ i ) 2 µ i N ν i µ i ) 2 )e λ m 2 M t for all t, µ i where λ 2 is the spectral gap of G. Moreover, Set T = M λ 2 m ρt) µ 2 M m µ ν 2 e λ 2 m M t for all t. 18M log ). One obtains m 3 ρt ) µ 2 m2 18 µ ν 2 m2 9 µ ρt ) 2 + ρt ) ν 2 ), which implies ρt ) µ m2 ρt ) ν 2 m2 4 as m 1 < 1. Thus ρt ) Nµ), where N Nµ) = {ρ = ρ i ) N M : ρ i µ i m, for i = 1, 2,, N} 2 is a compact convex subset of M with respect to Euclidean metric. In other words, after time T, the solution of 1.4) travels at least half of the distance from ν to µ and enters into the neighborhood Nµ). Then we can use the exactly same method as in Theorem 4.2 to estimate d 2 m, ). Denote λ 2 ρ) and λ N ρ) the second smallest eigenvalue and largest eigenvalue of LG, w m ρ)) respectively. Similar to the proof of Theorem 4.2, we can prove 1 4.7) λ N ρ) σ 2 gρ m σ, σ) 1 λ 2 ρ) σ 2 for any σ T ρ M. For ρ M, let δρ) be the maximal of the diagonal elements in the Laplacian matrix LG, w m ρ)) and let i w m ρ)g) = min wij m ρ)/ X ). X V, X N/2 i X,j X where the minimum is taken over all nonempty subsets X of V satisfying X N 2. We shall refer to i w m ρ)g) as the isoperimetric number of the weighted graph G, w m ρ)).

24 24 R. CHE, W. HUANG, Y. LI, AND P. TETALI Since G is connected and wij m ρ) min{ρ i : 1 i N} for {a, a j } E, it is not hard to see that 4.8) i w m ρ)g) 2 min{ρ i : 1 i N}. N It follow from Theorem 2.2 in [2] that the spectral gap λ 2 ρ) of the weighted graph G, w m ρ)) satisfies λ 2 ρ) δρ) δρ)2 i w m ρ)g) 2, It then follows from inequality δρ) δρ)2 i mρ)g) 2 w i w m ρ)g) 2 2 δρ) that 4.9) λ 2 ρ) 2min{ρ i : 1 i N}) 2 DN 2 by 4.8) and the fact that δρ) D. Let C 1 = max{ 1 λ 2 : ρ Nµ)}. Note that min{ρ ρ) i : 1 i N} m for all 2 ρ Nµ). We have C 1 2DN 2 by 4.9). It is well known that λ m 2 N ρ) N for ρ M see for example [2]). Let C 2 = inf{ 1 : ρ M}. Then C λ N ρ) 2 1 N have 4.1) C 2 σ 2 g m ρ σ, σ) for all σ T ρ M, ρ M and 4.11) g m ρ σ, σ) C 1 σ 2. Now by 4.7), we for all σ T ρ M, ρ Nµ). Moreover since the Fokker-Planck equation equation 1.4) is the generalized gradient flow of free energy F under the metric d Ψ, ) see Equation 45) and Theorem 3 in [6]), we have df ρt)) = g Ψ ρt) ρt), ρt)) for t >. By integrating the previous equality from to T, we have F ν) F ρt )) = T 1 T T g Ψ ρt) ρt), ρt)) 1 T T C2 ρt) ) 2 C 2 ν ρt ) 2 T the last second inequality comes from 4.1). At the same time, F ν) F ρt )) = the last inequality comes from 4.4). T 1 T T g Ψ ρt) ρt), ρt)) g Ψ ρt) ρt), ρt))) 2 1 T d2 Ψν, ρt )) 1 T d2 mν, ρt )) g Ψ ρt) ρt), ρt))) 2

25 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 25 Let st) = tρt ) + 1 t)µ for t [, 1]. Since Nµ) is a convex subset of R N and ρt ), µ Nµ). we have st) Nµ) for t [, 1]. Thus 1 d 2 mρt ), µ) gst) m ρt ) µ, ρt ) µ))2 1 C1 ρt ) µ 2 ) 2 = C 1 ρt ) µ 2 the last second inequality comes from 4.11) and the fact that st) Nµ). This gives us the bounds and d 2 mρt ), µ) C 1 ρt ) µ 2 m2 8 C 1 ρt ) ν 2 Finally, note that C 1 2DN 2 m 2 where K = MDN 3 +4) 2λ 2 m m2 T C 1 8C 2 F ν) F ρt ))) m2 T C 1 8C 2 F ν), d 2 mν, ρt )) T F ν) F ρt ))) T F ν). and C 2 1. We have N d 2 mν, µ) 2d 2 mν, ρt )) + 2d 2 mρt ), µ) m2 T C 1 + 2T )F ν) = T m2 C 1 + 2)F ν) 4C 2 4C 2 M 2DN 2 λ 2 m log18m m 3 )m2 m ) F ν) = KHν µ), 4 1 N log 18M m 3 ). This completes the proof. References [1] L. Ambrosio, N. Gigli and G. Savare, Gradient Flows: In Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich Birkhäuser Verlag, Basel, 28. [2] A. Berman and X.D. Zhang, Lower bounds for the eigenvalues of Laplacian matrices, Linear Algebra Appl., 3162), no.1-3, [3] S. Bobkov and F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., ), no.1, [4] S. Bobkov, I. Gentil and M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations, J. Math. Pures Appl., 821), no.7, [5] S. Bobkov and P. Tetali, Modified log-sobolev inequalities, mixing and hypercontractivity, J. Theoret. Probab., 1926), no.2, [6] S.N. Chow, W. Huang, Y. Li and H.M. Zhou, Fokker-Planck equations for a free energy functional of a Markov process on a graph, Arch. Ration. Mech. Anal., 23212), no.3, [7] M. Erbar and J. Maas, Ricci curvature of finite Markov chains via convexity of the entropy, Arch. Ration. Mech. Anal., ), no.3, [8] C.W. Gardiner, Handbook of stochastic methods. For physics, chemistry and the natural sciences. Third edition. Springer series in Synergetics, vol. 13, Springer-Verlag, Berlin, 24. [9] N. Gozlan, C. Roberto and P-M. Samson, P. Tetali, Displacement convexity of entropy and related inequalities on graphs, Probab. Th. Related Fields, August 213.

26 26 R. CHE, W. HUANG, Y. LI, AND P. TETALI [1] R. Jordan, D. Kinderlehrer and F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., ), no.1, [11] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. of Math. 2), 16929), no.3, [12] J. Lott and C. Villani, Hamilton-Jacobi semigroup on length spaces and applications, J. Math. Pures Appl., 8827), no.3, [13] M. Lu, L. Zhang and F. Tian, Lower bounds of the Laplacian spectrum of graphs based on diameter, Linear Algebra Appl., 4227), no.2-3, [14] D.H. Mathews and D.H. Turner, Prediction of RNA secondary structure by free energy minimization, Current Opinion in Structural Biology, 163)26), [15] J. Maas, Gradient flows of the entropy for finite Markov Chains, J. Funct. Anal., ), no.8, [16] A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24211), no.4, [17] F. Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations, 2621), no.1-2, [18] F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 1732), no.2, [19] H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications, Second edition, Springer Series in Synergetics, vol. 18, Springer-Verlag, Berlin, [2] O. Rojo, A nontrivial upper bound on the largest Laplacian eigenvalue of weighted graphs, Linear Algebra Appl., 4227), no.2-3, [21] M. Sammer, Aspects of mass transportation in discrete concentration inequalities. Thesis Ph.D.) Georgia Institute of Technology, 25, 119 pp. [22] M. Sammer and P. Tetali, Concentration on the discrete torus using transportation, Combin. Probab. Comput., 1829), no.5, [23] Z. Schuss, Singular perturbation methods in stochastic differential equations of mathematical physics, SIAM Rev., 22198), no.2, [24] K.T. Sturm, On the geometry of metric measure spaces I, Acta Math., 19626), no.1, [25] K.T. Sturm, On the geometry of metric measure spaces II, Acta Math., 19626), no.1, [26] B. Smolka and K. W. Wojciechowski, Contrast enhancement of badly illuminated images based on Gibbs distribution and random walk model, Lecture Notes in Computer Science, vol. 1296, Springer Berlin, 1997, [27] A. Torgasev and M. Petrovic, Lower bounds of the Laplacian graph eigenvalues, Indag. Mathem. N.S.), 1524), no.4, [28] C. Villani, Optimal Transport. Old and New, Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin, 29. [29] C. Villani, Topics in Optimal Transportation. Graduate Studies in Mathematics 58, American Mathematical Society, Providence 23). [3] Y. Wu, G. Hua and T. Yu, Tracking Articulated Body by Dynamic Markov Network, the Ninth IEEE International Conference on Computer Vision ICCV 23) 2-Volume Set.

27 CONVERGENCE TO EQUILIBRIUM AND TALAGRAND-TYPE INEQUALITIES 27 R. Che: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei Anhui 2326, PRC address: W. Huang: Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academy of Sciences, Hefei Anhui 2326, PRC address: Y. Li: Courant Institute of Mathematical Sciences, New York University, New York, NY 112, USA address: P. Tetali: School of Mathematics, Georgia Institute of Technology, Atlanta, GA 3332, USA address: