On the convexity of the entropy along entropic interpolations

Convexity properties of the entropy along displacement interpolations are crucial in the Lott-Sturm-Villani theory of lower bounded curvature of geodesic measure spaces. As discrete spaces fail to be geodesic, an alternate analogous theory is necessary in the discrete setting. Replacing displacement interpolations by entropic ones allows for developing a rigorous calculus, in contrast with Otto's informal calculus. When the underlying state space is a Riemannian manifold, we show that the first and second derivatives of the entropy as a function of time along entropic interpolations are expressed in terms of the standard Bakry-\'Emery operators $\Gamma$ and $ \Gamma_2$. On the other hand, in the discrete setting new operators appear. Our approach is probabilistic; it relies on the Markov property and time reversal. We illustrate these calculations by means of Brownian diffusions on manifolds and random walks on graphs. We also give a new unified proof, covering both the manifold and graph cases, of a logarithmic Sobolev inequality in connection with convergence to equilibrium.


Introduction
Displacement convexity of the relative entropy plays a crucial role in the Lott-Sturm-Villani (LSV) theory of curvature lower bounds of metric measure spaces [Stu06a,Stu06b,LV09,Vil09] and the related Ambrosio-Gigli-SavarÃľ gradient flow approach [AGS08,AGS].
Let us explain shortly what is meant by displacement convexity of the relative entropy. Let (X , d, m) be a metric measure space and P(X ) denote the space of all Borel probability measures on X . The distance d is the basic ingredient of the construction of the displacement interpolations and the positive measure m enters the game as being the reference measure of the relative entropy which is defined by H(µ|m) := X log(dµ/dm) dµ ∈ (∞, ∞], µ ∈ P(X ). Its displacement convexity means that along any displacement interpolation [µ 0 , µ 1 ] disp = (µ t ) 0≤t≤1 between two probability measures µ 0 and µ 1 , the entropy H(t) := H(µ t |m), 0 ≤ t ≤ 1 of the interpolation as a function of time admits some convexity lower bound. Let us recall briefly what displacement interpolations are. The Wasserstein pseudodistance W 2 is the square root of the optimal quadratic transport cost between µ 0 and µ 1 ∈ P(X ) given by W 2 2 (µ 0 , µ 1 ) := inf π X 2 d 2 dπ where the infimum is taken over all the couplings π ∈ P(X 2 ) of µ 0 and µ 1 . It becomes a distance on the subset P 2 (X ) of all µ ∈ P(X ) such that X d 2 (x o , x) µ(dx) < ∞, for some x o ∈ X . Displacement interpolations are the geodesics of the metric space (P 2 (X ), W 2 ). They were introduced in McCann's PhD thesis [McC94] together with the notion of displacement convexity of the entropy. Related convex inequalities turn out to be functional and geometric inequalities (Brunn-Minkowski, Prekopa-Leindler, Borell-Brascamp-Lieb, logarithmic Sobolev, Talagrand inequalities), see [McC97,CEMS01]. As a typical result, it is known [OV00, CEMS01, Sv05] that a Riemannian manifold has a nonnegative Ricci curvature if and only if t → H(t|vol) is convex along any displacement interpolation (µ t ) 0≤t≤1 .
An important hypothesis of the LSV theory is that the metric space is geodesic. This rules discrete spaces out. Something new must be found to develop an analogue of this theory in the discrete setting. Several attempts in this direction were proposed recently. Maas and Mielke [Maa11,Mie11,EM12,Mie] have discovered a Riemannian distance on the set of probability measures such that the evolution of continuous-time Markov chains on a graph are gradient flows of some entropy with respect to this distance. Bonciocat and Sturm in [BS09] and Gozlan, Roberto, Samson and Tetali in [GRST] proposed two different types of interpolations on the set of probability measures on a discrete metric measure graph which play the role of displacement interpolations in the LSV theory. The author recently proposed in [Léoa] a general procedure for constructing displacement interpolations on graphs by slowing down another type of interpolation, called entropic interpolation.
Entropic interpolations are the main actors of the present paper. Analogously to the displacement interpolations which solve dynamical transport problems, they solve dynamical entropy minimization problems. Although related to the purpose of this article, [Léoa] is mainly concerned by the slowing down asymptotic to build displacement, rather than entropic, interpolations. Displacement interpolations are connected to optimal transport, while entropic interpolations are connected to minimal entropy.
In this article, we consider the entropic interpolations in their own right, without slowing down. The goal remains the same: studying convexity of the entropy along these interpolations to derive curvature lower bounds of some Markov generators and state spaces. Our main result states that along any entropic interpolation [µ 0 , µ 1 ] = (µ t ) 0≤t≤1 between two probability measures µ 0 and µ 1 , the first and second derivatives of the relative entropy H(t) := H(µ t |m) of µ t with respect to some reference measure m can be written as The arrows on Θ and Θ 2 are related to the forward and backward directions of time.
The functions ϕ and ψ depend on the endpoints µ 0 and µ 1 , but the nonlinear operators − → Θ , ← − Θ , − → Θ 2 and ← − Θ 2 only depend on some reference m-stationary Markov process. For instance, when this process is the Brownian diffusion with generator L = (∆ − ∇V · ∇)/2 on a Riemannian manifold with m = e −V vol its reversing measure, we show that are respectively half the carrÃľ du champ Γu := L(u 2 ) − 2uLu and the iterated carrÃľ du champ Γ 2 u := LΓu−2Γ(u, Lu) which were introduced by Bakry and Émery in their article [BE85] on hypercontractive diffusion processes. In the discrete case, these operators are not linked to Γ and Γ 2 anymore, but they depend on L in a different manner, see (16) and (17). In the Riemannian case, Γ 2 is related to the Ricci curvature via Bochner's formula (34). This is the main connection between geometry and displacement convexity. Because of the tight relation (2) between (Θ, Θ 2 ) and (Γ, Γ 2 ) in the Riemannian case, it is natural to expect that (Θ, Θ 2 ) has also some geometric content in the discrete case. Does it allow for an efficient definition of curvature of graphs? This question is left open in the present paper. However, based on Θ and Θ 2 , we give a unified proof of the logarithmic Sobolev inequality on manifolds and a modified version on graphs. This is a clue in favor of the relevance of the entropic interpolations. An advantage of the entropic calculus, compared to its displacement analogue, is that entropic interpolations are regular in general. This is in contrast with displacement interpolations. Otto's informal calculus provides heuristics that necessitate to be rigorously proved by means of alternate methods while on the other hand entropic interpolations allow for a direct rigorous calculus. As a matter of fact, it is proved in [Léo12,Léoa] that displacement interpolations are semiclassical limits of entropic interpolations: entropic interpolations are regular approximations of displacement interpolations.
Another pleasant feature of the entropic approach is that it is not only available for reversible reference Markov dynamics, but also for stationary dynamics (in the reversible case, the time arrows on Θ and Θ 2 disappear).
The article's point of view is probabilistic. Its basic ingredients are measures on path spaces: typically, Brownian diffusions on manifolds and random walks on graphs. The Markov property plays a crucial role and we take advantage of its time symmetry. Recall that it states that conditionally on the knowledge of the present state, past and future events are independent. In particular, a time-reversed Markov process is still Markov. Time-reversal stands in the core of our approach (even when the reference stochastic process is assumed to be reversible); it explains the forward and backward arrows on Θ and Θ 2 . In contrast with analytic approaches, very few probabilistic attempts have been implemented to explore geometric and functional inequalities. In this respect, let us cite the contributions of Cattiaux [Cat04] and Fontbona and Jourdain [FJ] where stochastic calculus is central. In the present article, stochastic calculus is secondary. The symmetry of the Markov property allows us to proceed with basic measure theoretical technics.
Outline of the paper. The article is organized as follows. Entropic interpolations are discussed at Section 1. In particular, we describe their dynamics. This permits us to write their equation of motion and derive our main abstract result (1) at Section 2. This abstract result is exemplified with Brownian diffusion processes at Section 3 and random walks on a countable graph at Section 4. A unified proof of an entropy-entropy production inequality (logarithmic Sobolev inequality) in both the Riemannian and discrete graph settings, is given at Section 5 in connection with convergence to equilibrium. The stationary nonreversible case is investigated. At Section 6, we propose a heuristics based on thought experiments in order to grasp the link between entropic and displacement interpolations. Finally, we address several open questions related to curvature and entropic interpolations, keeping in mind the LSV theory of geodesic measure spaces as a guideline. This article is a preliminary step and these open questions should be seen as part of a program toward an analogue of the LSV theory that is available in a discrete setting.
Notation. For any measurable space Y , we denote by P(Y ) and M + (Y ) the sets of all probability measures and positive measures on Y . Let Ω = D([0, 1], X ) be the space of all right-continuous and left-limited paths from the time interval [0, 1] taking their values in a Polish state space X endowed with its Borel σ-field. The canonical process X = (X t ) 0≤t≤1 is defined by X t (ω) = ω t ∈ X for all 0 ≤ t ≤ 1 and ω = (ω t ) 0≤t≤1 ∈ Ω. As usual, Ω is endowed with the canonical σ-field σ(X). For any T ⊂ [0, 1] and any positive measure Q on Ω, we denote X T = (X t ) t∈T , Q T = (X T ) # Q, σ(X T ) is the σ-field generated by X T . In particular, for each t ∈ [0, 1], Q t = (X t ) # Q ∈ M + (X ) and Q 01 = (X 0 , X 1 ) # Q ∈ M + (X 2 ).

Entropic interpolation
Entropic interpolations are defined and their equations of motion are derived. The continuous and discrete space cases are both imbedded in the same abstract setting. This prepares next section where the first and second derivatives of H(t) := Hµ t |m) are calculated.
(f, g)-transform of a Markov process. The h-transform of a Markov process was introduced by Doob [Doo57] in 1957. It is revisited and slightly extended in two directions: (1) We consider a two-sided version of the h-transform, which we call (f, g)-transform, taking advantage of the invariance of the Markov property with respect to time reversal; (2) We extend the notion of Markov property to unbounded positive measures on Ω, having in mind two typical examples: • The reversible Wiener process on R n , i.e. the law of the Brownian motion with Lebesgue measure as its initial law; • The reversible simple random walk on a countable locally finite graph.
Assumptions on R. We specify some R ∈ M + (Ω) which we call the reference path measure and we assume that it is Markov and m-stationary 1 where m ∈ M + (X ) is a σ-finite positive measure on X . See Definitions A.1 and A.4 at the appendix Section A.
. The (f, g)-transform P ∈ P(Ω) of R associated with the couple of functions (f 0 , g 1 ) is defined by with X 2 f 0 (x)g 1 (y) R 01 (dxdy) = 1 for P to be a probability measure.
Remarks 1.2. Let us write some easy facts about (f, g)-transforms and h-transforms.
(a) Taking f 0 = ρ 0 and g 1 = 1 gives us P = ρ 0 (X 0 )R which is the path measure with initial marginal µ 0 = ρ 0 m ∈ P(X ) and the same forward Markov dynamics as R. (b) Symmetrically, choosing f 0 = 1 and g 1 = ρ 1 corresponds to P = ρ 1 (X 1 )R which is the path measure with final marginal µ 1 = ρ 1 m ∈ P(X ) and the same backward Markov dynamics as R.
where τ is a stopping time and h is such that P is a probability measure.
Let us denote the backward and forward Markov transition kernels of R by and for m-almost all z ∈ X , The relation between (f 0 , g 1 ) and the endpoint marginals where the density functions ρ 0 and ρ 1 are defined by µ 0 = ρ 0 m µ 1 = ρ 1 m . The system of equations (5) was exhibited by SchrÃűdinger in 1931 [Sch31] in connection with the entropy minimization problem (S dyn ) below.
Entropic interpolation. Let us introduce the main notion of this article.
Definition 1.3 (Entropic interpolation). Let P ∈ P(Ω) be the (f, g)-transform of R given by (3). Its flow of marginal measures µ t := (X t ) # P ∈ P(X ), 0 ≤ t ≤ 1, is called the R-entropic interpolation between µ 0 and µ 1 in P(X ) and is denoted by When the context is clear, we simply write [µ 0 , µ 1 ], dropping the superscript R.
Next theorem tells us that µ t is absolutely continuous with respect to the reference measure m on X . We denote by ρ t := dµ t /dm, 0 ≤ t ≤ 1, its density with respect to m.
Identities (5) extend to all 0 ≤ t ≤ 1 as next result shows.
Theorem 1.4. Let P ∈ P(Ω) be the (f, g)-transform of R given by (3). Then, P is Markov and for all Proof. Without getting into detail, the proof works as follows. As m = R t , . The Markov property of R at the last but one equality is essential in this proof. For more detail, see [Léo14,Thm. 3.4].
With (6), we see that an equivalent analytical definition of [µ 0 , µ 1 ] R is as follows.
(d) Note that in general, for any intermediate time 0 < t < 1, R xy t is not a Dirac mass. This is in contrast with the standard displacement interpolation [δ x , δ y ] disp whose typical form is δ γ xy t where γ xy is the constant speed geodesic between x and y when this geodesic is unique. Any entropic interpolation is a mixture of marginal flows of bridges. This is expressed at (62). For a proof, see [Léo14].
SchrÃűdinger problem. The notion of entropic interpolation is related to the entropy minimization problem (S dyn ) below. In order to state this problem properly, let us first recall an informal definition of the relative entropy H(p|r) := log(dp/dr) dp ∈ (−∞, ∞] of the probability measure p with respect to the reference σ-finite measure r; see the appendix Section A for further detail. The dynamical SchrÃűdinger problem associated with the reference path measure R ∈ M + (Ω) is H(P |R) → min; P ∈ P(Ω) : P 0 = µ 0 , P 1 = µ 1 (S dyn ) It consists of minimizing the relative entropy H(P |R) of the path probability measure P with respect to the reference path measure R subject to the constraint that P 0 = µ 0 and P 1 = µ 1 where µ 0 , µ 1 ∈ P(X ) are prescribed initial and final marginals. We shall need a little bit more than f 0 , g 1 ∈ L 2 (m) in the sequel. The following set of assumptions implies that some relative entropies are finite and will be invoked at Theorem 1.7 below.
Assumptions 1.6. In addition to f 0 , g 1 ∈ L 2 (m) and the normalization condition X 2 f 0 (x)g 1 (y) R 01 (dxdy) = 1, the functions f 0 and g 1 entering the definition of the (f, g)transform P of R given at (3) satisfy where log + h := 1 {h>1} log h, and as a convention 0 log 0 = 0.
Theorem 1.7. Under the Assumptions 1.6, the (f, g)-transform P of R which is defined at (3) is the unique solution of (S dyn ) where the prescribed constraints µ 0 = ρ 0 m and µ 1 = ρ 1 m are chosen to satisfy (5).
Note that the solution of (S dyn ) may not be a (f, g)-transform of R. More detail about the SchrÃűdinger problem can be found in the survey paper [Léo14].
[µ, µ] R is not constant in time. One must be careful when employing the term interpolation as regards entropic interpolations, because in general when µ 0 = µ 1 =: µ, the interpolation [µ, µ] R is not constant in time. Let us give two examples. (a) Suppose that R 01 (x, x) > 0. Then the solution of (S dyn ) with µ 0 = µ 1 = δ x is the bridge R xx and [δ x , δ x ] R is the marginal flow (R xx t ) 0≤t≤1 which is not constant in general. (b) Consider R to be the reversible Brownian motion on the manifold X . Starting from µ at time t = 0, the entropic minimizer P is such that µ t := P t gets closer to the invariant volume measure m = vol on the time interval Forward and backward stochastic derivatives of a Markov measure. Since P is Markov, its dynamics is characterized by either its forward stochastic derivative and its initial marginal µ 0 or its backward stochastic derivative and its final marginal µ 1 . Before computing these derivatives, let us recall some basic facts about these notions. Let Q ∈ M + (Ω) be a Markov measure. Its forward stochastic derivative ∂ + − → L Q is defined by for any measurable function u : [0, 1] × X → R in the set dom − → L Q for which this limit exists Q t -a.e. for all 0 ≤ t < 1. In fact this definition is only approximate, we give it here as a support for understanding the relations between the forward and backward derivatives. For a precise statement see [Léoc,§2]. Since the time reversed Q * of Q is still Markov, Q admits a backward stochastic derivative −∂ + ← − L Q which is defined by for any measurable function u : [0, 1] × X → R in the set dom ← − L Q for which this limit exists Q t -a.e. for all 0 < t ≤ 1. Remark that ← − [Léoc,§2] that these stochastic derivatives are extensions of the extended forward and backward generators of Q in the semimartingale sense, see [DM87]. In particular, they offer us a natural way for computing the generators.
Forward and backward dynamics of the entropic interpolation. The dynamics in both directions of time of [µ 0 , µ 1 ] R are specified by the stochastic derivatives − → A := − → L P and ← − A := ← − L P of the Markov measure P ∈ P(Ω) which appears at Definition 1.3. For simplicity, we also denote , meaning that the forward and backward transition mechanisms of R do not depend on t.
To derive the expressions of − → A and ← − A , we need to introduce the carrÃľ du champ Γ of both R and its time reversed R * . The exact definition of the extended carrÃľ du champ Γ and its domain is given at [Léoc,Def. 4.10]. In restriction to the functions u, v on X such that u, v and uv are in dom − → L , we recover the standard definition which corresponds to an SDE driven by a standard Brownian motion. Consequently, In general, even if R is a reversible path measure, the prescribed marginal constraints enforce a time-inhomogeneous transition mechanism: the forward and backward deriva- It is known (see [Léoc] for instance) that for any function u : X → R belonging to some class of regular functions to be made precise, where f and g are defined at (4). Because of (6), for any t no division by zero occurs µ t -a.e. For (10) to be meaningful, it is necessary that the functions f and g are regular enough for f t and g t to be in the domains of the carrÃľ du champ operators. In the remainder of the paper, we shall only be concerned with Brownian diffusion processes and random walks for which the class of regular functions will be specified, see Sections 3 and 4.
Going back to (4), we see that the processes f t (X t ) and g t (X t ) are respectively backward and forward local R-martingales. In terms of stochastic derivatives, this is equivalent to where these identities hold µ t -almost everywhere for almost all t.
Assumptions 1.9. We assume that the kernels − → r , ← − r that appear in (4) are (a) positivity improving and (b) regularizing: The functions (f 0 , g 1 ) and the kernels − → r , ← − r are such that (t, z) → f (t, z), g(t, z) are twice t-differentiable classical solutions of the parabolic PDEs (11).
Assumption (a) implies that for any 0 < t < 1, f t and g t are positive everywhere. In particular, we see with (6) that µ t ∼ m. Assumption (b) will be made precise later in specific settings. It will be used when computing time derivatives of t → µ t .
Mainly because we are going to study the relative entropy H(µ t |m) = X log(f t g t ) dµ t , it is sometimes worthwhile to express − → A and ← − A in terms of the logarithms of f and g : In analogy with the Kantorovich potentials which appear in the optimal transport theory, we call ϕ and ψ the SchrÃűdinger potentials. Under Assumption 1.9-(b), they are classical solutions of the "second order" Hamilton-Jacobi-Bellman (HJB) equations where the non-linear operators which permits us to rewrite (10) as A ϕt , emphasizing their dependence on the SchrÃűdinger potentials.
Forward-backward systems. The complete dynamics of [µ 0 , µ 1 ] R is described by a forwardbackward system. We see that and A * is the algebraic adjoint of A : X Au dµ =: u, A * µ . The evolution equation for µ is understood in the weak sense, in duality with respect to a large enough class of regular functions u. Similarly, It appears that the boundary data for the systems (14) and (15) are (µ 0 , g 1 ) and (f 0 , µ 1 ).

Second derivative of the entropy
The aim of this section is to provide basic formulas to study the convexity of the entropy (5). We have also seen at (14) and (15) that it is specified by either (µ 0 , g 1 ) or (f 0 , µ 1 ).
Basic rules of calculus and notation. We are going to use the subsequent rules of calculus and notation where we drop the subscript t as often as possible. Assumption 1.9 is used in a significant way: it allows to work with everywhere defined derivatives.
The first identity is Theorem 1.4. The others are notation which are well defined everywhere under Assumption 1.9. •u := ∂ t u,μ := ∂ t µ, u, η := X u dη. These are simplifying notation.
• The first line of the following equalities is (10) and the other ones are definitions: These identities hold since µ is the time-marginal flow of the Markov law P whose forward and backward derivatives are − → A and ← − A . • A short way for writing (11) and (13) is: Entropy production. By Assumption 1.9, the evolution PDEs (11) and (13) are defined in the classical sense and the functions − → I and ← − I below are well defined.
Definition 2.1 (Forward and backward entropy production). For each 0 ≤ t ≤ 1, we define respectively the forward and backward entropy production at time t along the inter- where for any regular enough function u, Calling the functions − → I and ← − I "entropy production" is justified by Corollary 2.4 below. We shall see at Section 3 that in the reversible Brownian diffusion setting where R is associated with the generator (9), we have with no dependence on t and as usual, ones simply writes Γ(u) := Γ(u, u).
Proposition 2.2. Suppose that the Assumptions 1.6 and 1.9 hold. The first derivative Proof. With Theorem 1.4 and the definition of the relative entropy, we immediately see that H(t) = log ρ, µ = ϕ + ψ, µ . It follows with our basic rules of calculus that The point here is to apply the forward operators As an immediate consequence of this Proposition 2.2, we obtain the following Corollary 2.4 about the dynamics of heat flows.  . It corresponds to f 0 = 1 and g 1 = ρ 1 , i.e. to P = ρ 1 (X 1 ) R.
(a) Along any forward heat flow (µ t ) 0≤t≤1 , we have: Second derivative. The computations in this subsection are informal and the underlying regularity hypotheses are kept fuzzy. We assume that the Markov measure R is nice enough for the SchrÃűdinger potentials ϕ and ψ to be in the domains of the compositions of the operators L, Γ, A, B and C which are going to appear below. To keep formulas into a reasonable size, we have assumed that R is m-stationary. The informal results that are stated below at Claims 2.6 and 2.7 will turn later at Sections 3 and 4 into formal statements in specific settings. We introduce provided that the function u is such that these expressions are well defined. It will be shown at Section 3, see (30), that in the Brownian diffusion setting where R is associated are the forward and backward iterated carrÃľ du champ operators.
Remark 2.5. We keep the notation of Remark 1.8. In the standard Bakry-Émery setting, the iterated carrÃľ du champ of In the present paper, we consider L = L/2 instead of L and we have already checked at Remark 1.8 that Γ = Γ. Consequently, Γ 2 = Γ 2 .
Introducing these definitions is justified by the following claim.
Claim 2.6 (Informal result). Assume that R is m-stationary. Then, Similarly, we obtain which completes the proof of the claim.
Gathering Proposition 2.2 and Claim 2.6, we obtain the following

Brownian diffusion process
As a first step, we compute informally the operators Θ and Θ 2 associated with the Brownian diffusion process on X = R n whose forward and backward generators are given for all Here, z → − → b (z), ← − b (z) ∈ R n are the forward and backward drift vector fields. The term "informally" means that we suppose that − → b and ← − b satisfy some unstated growth and regularity properties which ensure the existence of R and also that Assumptions 1.9 are satisfied. Then, we consider reversible Brownian diffusion processes on a compact manifold.
Dynamics of the entropic interpolations. The associated nonlinear operators are of the forward and backward derivatives tell us that the density µ t (z) := dµ t /dz solves the following forward-backward system of parabolic PDEs where ψ solves the HJB equation In the reversed sense of time, we obtain where ϕ solves the HJB equaltion We put log g 1 = −∞ on the set where g 1 vanishes and the boundary condition at time t = 1 must be understood as lim t↑1 ψ t = log g 1 . Similarly, we have also lim Because of the stochastic representation formulas (4), the functions ϕ and ψ are the unique viscosity solutions of the above Hamilton-Jacobi-Bellman equations, see [FS93, Thm. II.5.1]. The existence of these solutions is ensured by the Assumptions 1.6.
are respectively the osmotic and the (forward) current velocities of R that were introduced by E. Nelson in [Nel67]. In fact, the first equation in (23) is satisfied in a stronger way, since the duality formula associated to time reversal [Föl86] is An interesting situation is given by It is easily seen that is the stationary measure of R. Moreover, with (24), we also obtain In dimension 2, choosing b ⊥ = e V (−∂ y U, ∂ x U) with U : R 2 → R a C 2 -regular function, solves (25). Regardless of the dimension, b ⊥ = e V S∇U where S is a constant skewsymmetric matrix and U : R n → R a C 2 -regular function, is also a possible choice. In dimension 3 one can take b ⊥ = e V ∇ ∧ A where A : R 3 → R 3 is a C 2 -regular vector field.
Computing Θ and Θ 2 . Our aim is to compute the operators − → Θ , ← − Θ , − → Θ 2 and ← − Θ of the Brownian diffusion process R with the forward and backward derivatives given by (20) and (26): where ∇ · (e V b ⊥ ) = 0. For simplicity, we drop the arrows and the index t during the intermediate computations.
Computation of Θ 2 . Since R is a diffusion, Γ is a derivation i.e. Γ(uv, w) = uΓ(v, w) + vΓ(u, w), for any regular enough functions u, v and w. In particular, the two last terms in the expression of Θ 2 u simplify: They are precisely half the iterated carrÃľ du champ operators − → Γ 2 and ← − Γ 2 defined at (18). The iterated carrÃľ du champ Γ o where ∇ 2 u is the Hessian of u and ∇ 2 u 2 HS = tr ((∇ 2 u) 2 ) is its squared Hilbert-Schmidt norm. As Regularity problems. To make Claim 2.7 a rigorous statement, one needs to rely upon regularity hypotheses such as Assumptions 1.9. If one knows that f 0 , g 1 , − → b and ← − b are such that the linear parabolic equations (11) admit positive C 2,2 ((0, 1)×X )-regular solutions then we are done. Verifying this is a standard (rather difficult) problem which is solved under various hypotheses. Rather than giving detail about this PDE problem which can be solved by means of Malliavin calculus, we present an example at next subsection.
Working with classical solutions is probably too demanding. The operators Θ and Θ 2 are functions of ∇u and the notion of gradient can be extended as in [Léoc] in connection with the notion of extended stochastic derivatives, see Remark 3.1-(a). Furthermore, when working with integrals with respect to time, for instance when considering integrals of the entropy production (see Section 5), or in situations where H(t) is known to be twice differentiable almost everywhere (e.g. H(t) is the sum of a twice differentiable function and a convex function), it would be enough to consider dt-almost everywhere defined extended gradients. This program is not initiated in the present paper.
Reversible Brownian diffusion process on a Riemannian manifold X . We give some detail about the standard Bakry-Émery setting that already appeared at Remark 1.8. It corresponds to the case where b ⊥ = 0. Let us take where V ∈ C 2 satisfies X e −V (x) vol(dx) < ∞. The m-reversible Brownian diffusion process R ∈ M + (Ω) is the Markov measure with the initial (reversing) measure m and the semigroup generator L = (−∇V · ∇ + ∆)/2. The reversible Brownian motion R o ∈ M + (Ω) corresponds to V = 0. Its stochastic derivatives are and vol is its reversing measure. Since R o is the unique solution to its own martingale problem, there is a unique R which is both absolutely continuous with respect to R o and m-reversible.
Lemma 3.2. In fact R is specified by Proof. To see this, let us defineR by means of dR/dR o := e −V (X 0 ) Z 1 where On the other hand, since both R o and dR/dR o are invariant with respect to the time reversal X * : X * t := X 1−t, t∈[0,1] ,R is also invariant with respect to time reversal: (X * ) #R =R. In particular, its endpoint marginals are equal:R 0 =R 1 . Consequently,R [0,t] doesn't send mass to a cemetery point † outside X as time increases, for otherwise its terminal marginalR 1 would give a positive mass to †, in contradiction withR 0 ( †) = 0 and R 0 =R 1 . Hence, Z is a genuine forward R o -martingale. With Itô's formula, we see that dZ t = −Z t ∇ V 2 (X t ) · dX t , R o -a.e. and by Girsanov's theory we know thatR is a (unique) solution to the martingale problem associated with the generator L = (−∇V · ∇ + ∆)/2. Finally, we take R =R and it is easy to check that L is symmetric in L 2 (m), which implies that m is reversing.
Remark 3.3. When L is given by (33), for any non-zero nonnegative functions f 0 , g 1 ∈ L 2 (vol), the smoothing effect of the heat kernels − → r and ← − r in (4) allows us to define classical gradients ∇ψ t and ∇ϕ t for all t in [0, 1) and (0, 1] respectively. We see that ∇ψ t and ∇ϕ t are the forward and backward drift vector fields of the canonical process under P.
Next result proposes a general setting where Assumptions 1.9 are satisfied. The manifold X is assumed to be compact to avoid integrability troubles.
Proposition 3.4. Suppose that X is a compact Riemannian manifold without boundary and V : X → R is C 4 -regular. Then, for any C 2 -regular function u : X → R, the function (t, x) → u(t, x) := E R [u(X t ) | X 0 = x] belongs to C 1,2 ((0, 1) × X ).
In particular, if in addition X is assumed to be connected and f 0 and g 1 are non-zero nonnegative C 2 -regular functions, the functions f and g defined at (4) and their logarithms ϕ and ψ are classical solutions of (11) and (13).
Proof. For any path ω ∈ Ω and any 0 ≤ t ≤ 1, we denote With Lemma 3.2 we see that for all where (B x s ) 0≤s≤1 is a Brownian motion starting from x under some abstract probability measure whose expectation is denoted by E. By means of parallel transport, it is possible to build on any small enough neighborhood U of x a coupling such that x ′ ∈ U → B x ′ is almost surely continuous with respect to the uniform topology on Ω. This coupling corresponds to B x = x + B 0 in the Euclidean case. The announced x-regularity is a consequence of our assumptions which allow us to differentiate (in the usual deterministic sense) in the variable x under the expectation sign E. On the other hand, the t-regularity is a consequence of stochastic differentiation: apply ItÃť's formula to u(B x t ) and take advantage of the martingale property of Z t (B x ).
As regards last statement, the connectivity assumption implies the positivity of f and g on (0, 1) × X as soon as f 0 and g 1 are nonnegative and not vanishing everywhere.
We gave the detail of the proof of Proposition 3.4 because of its ease. But in view of Remark 3.3, the requirement that f 0 and g 1 are C 2 is not optimal. However, this restriction will not be harmful when investigating convexity properties of the entropy along interpolations.
In view of Proposition 3.4, we remark that under its assumptions the functions f t , g t belong to the domain of the carrÃľ du champ operator. Hence, for any 0 < t < 1, the stochastic derivatives of P are well-defined on C 2 (X ) and equal to − Here, ψ and ϕ are respectively the classical solutions (compare Remark 3.1-(a)) of the HJB equations (21) and (22) with Γ o 2 (u) = ∇ 2 u 2 HS + Ric(∇u) (34) where ∇ 2 u is the Hessian of u and ∇ 2 u 2 HS = tr ((∇ 2 u) 2 ) is its squared Hilbert-Schmidt norm. As Theorem 3.5 (Reversible Brownian diffusion process). Let the reference Markov measure R be associated with L given at (9) on a compact connected Riemannian manifold X without boundary. It is assumed that V : X → R is C 4 -regular and f 0 , g 1 are nonzero nonnegative C 2 -regular functions. Then, along the entropic interpolation [µ 0 , µ 1 ] R associated with R, f 0 and g 1 , we have for all 0 < t < 1, Proof. The assumptions imply that f 0 and g 1 satisfy (8) and they allow us to apply Proposition 3.4 so that Assumptions 1.9 are satisfied and Claim 2.7 is a rigorous result.

Random walk on a graph
Now we take as our reference measure R a continuous-time Markov process on a countable state space X with a graph structure (X , ∼). The set X of all vertices is equipped with the graph relation x ∼ y which signifies that x and y are adjacent, i.e. {x, y} is a non-directed edge. The degree n x := # {y ∈ X , x ∼ y} of each x ∈ X is the number of its neighbors. It is assumed that (X , ∼) is locally finite, i.e. n x < ∞ for all x ∈ X and also that (X , ∼) is connected. This means that for any couple (x, y) ∈ X 2 of different states, there exists a finite chain (z 1 , . . . , z k ) in X such that x ∼ z 1 ∼ z 2 · · · ∼ z k ∼ y. In particular n x ≥ 1, for all x ∈ X .

Dynamics of the entropic interpolation. A (time homogeneous) random walk on
is the instantaneous frequency of forward jumps from x to y, is the discrete gradient and is the forward jump kernel. Similarly, one denotes its backward derivative by For the m-stationary Markov measure R with stochastic derivatives − → L and ← − L , the timereversal duality formula is The expressions − → L u and ← − L u can be seen as the matrices acting on the column vector u = [u x ] x∈X . Therefore, the solutions f and g of (11) are ∈ R X are column vectors, whenever these exponential matrices are well-defined. Let us compute the operators B, Γ and A. By a direct computation, we obtain for all

The matrix representation of these operators is
The forward-backward systems describing the evolution of [µ 0 , µ 1 ] R are where the measures µ t are seen as row vectors and the functions f 0 , g 1 as column vectors.
Derivatives of the entropy. A pleasant feature of the discrete setting is that no spatial regularity is required for a function to be in the domain of the operators L, Γ, . . . The only required regularity is that the functions f and g defined by (4) have to be twice differentiable with respect to t on the open interval (0, 1). But this is ensured by the following lemma.
Lemma 4.1. Consider f 0 , g 1 as in Definition 1.1 and suppose that f 0 , g 1 ∈ L 1 (m)∩L 2 (m) where the stationary measure m charges every point of X . Then, under the assumption that sup for each x ∈ X , t → f (t, x) and t → g(t, x) are C ∞ -regular on (0, 1).
Proof. For any 0 ≤ t ≤ 1, f t and g t are well-defined P t -a.e., where P is given by (3). But, as R is an irreducible random walk, for each 0 < t < 1, "P t -a.e." is equivalent to "everywhere". Since f 0 ∈ L 1 (m) and R is m-stationary, we have As m charges every point, we also see that t ∈ (0, 1) → f t (x) ∈ R is infinitely differentiable for every x. A similar proof works with g instead of f .
As an important consequence of Lemma 4.1 for our purpose, we see that the statement of Claim 2.7 is rigorous in the present discrete setting.
Theorem 4.2. Let R be an m-stationary random walk with jump measures − → J and ← − J which satisfy (37). Along any entropic interpolation [µ 0 , µ 1 ] R associated with a couple (f 0 , g 1 ) as in Definition 1.1 and such that f 0 , g 1 ∈ L 1 (m)∩L 2 (m), we have for all 0 < t < 1, where the expressions of − → Θ ψ t and ← − Θ ϕ t are given at (38) and the expressions of − → Θ 2 ψ t and ← − Θ 2 ϕ t are given at Proposition 4.4 below.
Computing Θ and Θ 2 . Our aim now is to compute the operators − → Θ , ← − Θ , − → Θ 2 and ← − Θ 2 for a general random walk R. We use the shorthand notation X a(x, y) J x (dy) = [ X a dJ](x) and drop the arrows and the index t during the intermediate computations. The Hamilton-Jacobi operator is Bu := e −u Le u = X (e Du − 1) dJ which gives where the function θ is defined by θ(a) := e a − a − 1, a ∈ R. Compare Cu = |∇u| 2 /2, remarking that θ(a) = a 2 /2 + o a→0 (a). The convex conjugate θ * of θ will be used in a moment. It is given by where we used e Dψt(x,y) − 1 = Dg t (x, y)/g t (x) and e Dϕt(x,y) − 1 = Df t (x, y)/f t (x). The ratio Dg/g should be seen as the discrete logarithmic derivative of g. Recall that g t > 0, µ t -a.e. : we do not divide by zero. Compare Θψ = |∇g/g| 2 /2, remarking that θ * Entropy production. The entropy productions are Computation of Θ 2 . Unlike the diffusion case, no welcome cancellations occur. The carré du champ Γ is not a derivation anymore since Γ(uv, w) − [uΓ(v, w) + vΓ(u, w)] = uvLw + X DuDvDw dJ. Furthermore, we also loose the simplifying identity C = Θ. To give a readable expression of Θ 2 , it is necessary to introduce some simplifying shorthand notation: , y), Du(y, z)) J x (dy)J y (dz) x→y→z F (a, c) = X 2 F (Du(x, y), Du(x, z)) J x (dy)J y (dz).
Remark that x→y→z F (c) = X F (Du(x, z)) J 2 x (dz) with J 2 x (dz) := y∈X J x (dy)J y (dz) and a + b = c.
Proof. The first identity is (38). Let us look at Θ 2 u(x). Rather than formula (17), for an explicit formulation of Θ 2 in terms of J, it will be easier to go back to (19): − 1), we see that The functions defining the integrands rewrite as follows

Hence,
are the forward and backward jump frequencies of P = f 0 (X 0 )g 1 (X 1 ) R.
Reversible random walks. Let us take a positive measure m = x∈X m x δ x ∈ M + (X ) with m x > 0 for all x ∈ X . It is easily checked with the detailed balance condition which characterizes the reversibility of m that the jump kernel where s(x, y) = s(y, x) > 0 for all x ∼ y, admits m as a reversing measure. As (X , ∼) is assumed to be connected, the random walk is irreducible and the reversing measure is unique up to a multiplicative constant. (a) Reversible counting random walk. The simplest example is provided by the counting jump kernel J x = y:x∼y δ y ∈ P(X ), x ∈ X which admits the counting measure m = x δ x as a reversing measure: take m x = 1 and s(x, y) = 1 in (44). (b) Reversible simple random walk. The simple jump kernel is J x = 1 nx y:x∼y δ y ∈ P(X ), x ∈ X . The measure m = x n x δ x is a reversing measure: take m x = n x and s(x, y) = (n x n y ) −1/2 in (44).
The dynamics of R is as follows. Once the walker is at x, one starts an exponential random clock with frequency J x (X ). When the clock rings, the walker jumps at random onto a neighbor of x according to the probability law J x (X ) −1 y:x∼y J(x; y) δ y . This procedure goes on and on. Since R is reversible, we have − → J = ← − J = J and the forward and backward frequencies of jumps of the (f, g)-transform P = f 0 (X 0 )g 1 (X 1 ) R are respectively As a direct consequence of Theorem 4.2 and Proposition 4.4, we obtain the following result.
Theorem 4.6. Let R be an m-reversible random walk with jump measure J which is given by (44) and satisfies sup x∈X J x (X ) < ∞.
For this condition to be satisfied, it suffices that for some 0 < c, σ < ∞, we have for all with the notation of (44). Then, along any entropic interpolation [µ 0 , µ 1 ] R associated with a couple (f 0 , g 1 ) as in Definition 1.1 and such that f 0 , g 1 ∈ L 1 (m) ∩ L 2 (m), we have for all 0 < t < 1, where the expressions of Θψ t and Θϕ t are given at (38) and the expressions of Θ 2 ψ t and Θ 2 ϕ t are given at Proposition 4.4 (drop the useless time arrows).

Convergence to equilibrium
Let R be an m-stationary Markov probability measure. In this section m is assumed to be a probability measure (and so is R). We are interested in the convergence as t tends to infinity of the forward heat flow where ρ 0 = dµ 0 /dm is the initial density. If the solution of the forward heat equation with the initial profile ν o = µ 0 is unique, then it is the heat flow (46).
But in general (µ t ) t≥0 is only a solution of this heat equation. We are going to prove by implementing the "stochastic process strategy" of the present paper, that under some hypotheses, the following convergence to equilibrium lim t→∞ µ t = m holds. As usual, the function t → H(µ t |m) is an efficient Lyapunov function of the dynamics.
Stationary dynamics. Let us first make the basic assumptions precise.
Assumptions 5.1. The stationary measure m is a probability measure and we assume the following. (a) Brownian diffusion. The forward derivative is on a connected Riemannian manifold X without boundary. The initial density ρ 0 := dµ 0 /dm is regular enough for ρ t (z) = dµ t /dm(z) to be positive and twice differentiable in t and z on (0, ∞) × X .
on the countable locally finite connected graph (X , ∼). We assume that (37) holds: The point in the following Theorem 5.2 is that no curvature bound such as (49) below nor reversibility are required. We define We have seen at (28) and (38) that ← − Θ ≥ 0 (remark that θ * ≥ 0), this shows that I(µ|m) ≥ 0.
Proof. The main idea of this proof is to consider the forward heat flow (µ t ) t≥0 as an entropic interpolation, see Definition 2.3, and to apply Claim 2.7. It will be seen that the general regularity hypotheses of the theorem ensure that this claim turns into a rigorous statement.
Instead of restricting time to [0, 1], we allow it to be in [0, T ], T > 0, and let T tend to infinity. We denote ρ t := dµ t /dm. Since µ is the time-marginal flow of P = ρ 0 (X 0 ) R, we have f 0 = ρ 0 and g T = 1. This implies that g t = 1 and f t = ρ t . Indeed, for all 0 ≤ t ≤ T . Our assumptions guarantee that for all t > 0, ϕ t is regular enough to use our previous results about "Θ 2 -calculus". Denoting H(t) = H(µ t |m), with Corollary 2.4, we see that As I(·|m) ≥ 0, H is decreasing. Of course, H(0) − H(T ) = T 0 I(µ t |m) dt. Now, we let T tend to infinity. As H is non-negative and decreasing, it admits a limit H(∞) := lim T →∞ H(T ) and the integral in is convergent, implying lim t→∞ I(µ t |m) = 0.
Diffusion setting. By (29) with ϕ t = ρ t , this limit implies that lim t→∞ X |∇ √ ρ t | 2 dvol = 0. By PoincarÃľ's inequality, for any open bounded connected domain U with a smooth boundary, there exists a constant C U such that which in turns implies that lim t→∞ ρ t = c everywhere on X (recall that ρ is continuous on (0, ∞) × X and X is connected) for some constant c ≥ 0. Therefore, lim t→∞ ρ t = 1 everywhere. As ρ 0 is assumed to be bounded, for all t we have ρ t (X t ) = E R (ρ 0 | X t ) ≤ sup ρ 0 < ∞. This allows us to apply Lebesgue dominated convergence theorem to assert that lim t→∞ H(µ t |m) = 0.
Random walk setting. Since lim t→∞ I(µ t |m) = 0, we obtain lim t→∞ (x,y):x∼y We have for some constant 2 C > 0. In these three cases, we see that the convergence to zero of the left-hand side implies the convergence of b to a. The remaining case when b/a ≥ 2 and a ≤ 2 is controlled by considering the symmetric term bθ * (a/b − 1) which also appears in the series in the limit (48), reverting x and y; recall that ← − J x (y) > 0 ⇔ ← − J y (x) > 0. As bθ * (a/b − 1) ≥ b ≥ 2a ≥ 0, the convergence of the left-hand term to zero also implies the convergence of b to a. Therefore, the limit (48) implies that for all adjacent x and y ∈ X , lim t→∞ ρ t (x) = lim t→∞ ρ t (y). Since the graph is connected, we have lim t→∞ ρ t = 1 everywhere. Finally, the estimate sup t,x ρ t (x) ≤ sup X ρ 0 < ∞ allows us to conclude with the dominated convergence theorem.
holds for any nonnegative function ρ : X → [0, ∞) such that X ρ dm = 1, which in the diffusion setting (a) is also assumed to be C 2 -regular.
Proof of Theorem 5.4. We already saw during the proof of Theorem 5.2 that our general assumptions allow us to apply Θ 2 -calculus: Claim 2.7 is rigorous with ϕ t = log ρ t . This gives: , for all t ≥ 0. Integrating leads to ← − I (t) ≤ ← − I (0)e −κt , t ≥ 0, which is (50). Let us assume for a while, in addition to (49), that sup X ρ 0 < ∞. It is proved at Theorem 5.2 that under this restriction, H(∞) = 0. With (47) and (50), we see that A standard approximation argument based on the sequence ρ n = (ρ ∧ n)/ X (ρ ∧ n) dm and Fatou's lemma: H(ρm|m) ≤ lim inf n H(ρ n m|m), allows to extend this inequality to unbounded ρ. This proves (52).
Remark 5.5 (About time reversal). It appears with (49) that time reversal is tightly related to this convergence to stationarity. Let us propose an informal interpretation of this phenomenon. The forward entropy production − → I (t) vanishes along the forward heat flow (µ t ) t≥0 since − → I (t) = X − → Θ ψ t dµ t and ψ = 0. Similarly X − → Θ 2 ψ t dµ t = 0. No work is needed to drift along the heat flow. In order to evaluate the rate of convergence, one must measure the strength of the drift toward equilibrium. To do so "one has to face the wind" and measure the work needed to reach µ 0 when starting from m, reversing time.
(c) An inspection of the proof of Theorem 5.4 shows that only the integrated version of (49): is sufficient.
In the diffusion setting (27) on X = R n , we know with (31) that (49) becomes ≥ κ|∇u| 2 for any sufficiently regular function u. This Γ 2 -criterion was obtained by Arnold, Carlen and Ju [ACJ08]; see also the paper [HHMS05] by Hwang, Hwang-Ma and Sheu for a related result. These results are recovered in the recent paper [FJ] by Fontbona and Jourdain who implement a stochastic process approach, slightly different from the present article's one but where time reversal plays also a crucial role, see Remark 5.5.
Reversible dynamics. More precisely, we are concerned with the following already encountered m-reversible generators L. Assumptions 5.7. It is required that the reversing measure m is a probability measure. The assumptions (a) and (b) below allow us to apply respectively Theorems 3.5 and 4.6. (a) The Riemannian manifold X is compact connected without boundary and L is given at (54). We assume that V : X → R is C 4 -regular and without loss of generality that V is normalized by an additive constant such that m = e −V vol is a probability measure. (b) The countable graph (X , ∼) is assumed to be locally finite and connected. The generator L is given at (55) and the equilibrium probability measure m and the function s satisfy (45).
Reversibility allows for a simplified expression of I(·|m). Indeed, as ρΘ(log ρ) = L(ρ log ρ− ρ) − log ρ Lρ, we obtain where the last equality follows from the symmetry of L in L 2 (m) (a consequence of the m-reversibility of R), whenever (ρ, log ρ) ∈ dom Γ. Therefore, It is the Fisher information of µ with respect to m. A direct computation shows that (a) In the diffusion setting (a), (b) In the random walk setting (b),
In the diffusion setting (a), Theorem 5.8 is covered by the well-known result by Bakry and Émery [BE85]. Inequality (58) is the standard logarithmic Sobolev inequality and (57) is the usual Γ 2 -criterion: Γ 2 ≥ κΓ. In the random walk setting (b), inequality (58) is a modified logarithmic Sobolev inequality which was introduced by Bobkov and Tetali [BT06] in a general setting and was already employed by Dai Pra, Paganoni and Posta [DPPP02] in the context of Gibbs measures on Z d . Later Caputo, Dai Pra and Posta [CP07,CDPP09,DPP13] have derived explicit criteria for (58) to be satisfied in specific settings related to Gibbs measures of particle systems. These criteria are close in spirit to (57).

Some open questions about curvature and entropic interpolations
We have seen at Section 5 that convergence to equilibrium is obtained by considering heat flows. The latter are the simplest entropic interpolations since f = 1 or g = 1. In the present section, general entropic interpolations are needed to explore curvature properties of Markov generators and their underlying state space X by means of the main result of the article: Convexity properties of the entropy along displacement interpolations are fundamental for the Lott-Sturm-Villani (LSV) theory of lower bounded curvature of metric measure spaces [Stu06a,Stu06b,LV09,Vil09]. An alternate approach would be to replace displacement interpolations by entropic interpolations, taking advantage of the analogy between these two types of interpolations. Although this program is interesting in itself, it can be further motivated by remarking the following facts.
As regards (1), note that the dynamics of an entropic interpolation shares the regularity of the solutions f and g of the backward and forward "heat equations" (11). Their logarithms, the SchrÃűdinger potentials ϕ and ψ, solve second order Hamilton-Jacobi-Bellman equations. This is in contrast with a displacement interpolation whose dynamics is driven by the Kantorovich potentials (analogues of ϕ and ψ) which are solutions of first order Hamilton-Jacobi equations. Entropic interpolations are linked to regularizing dissipative PDEs, while displacement interpolations are linked to transport PDEs. More detail about these relations is given in [Léo14].
In practice, once the regularity of the solutions of the dissipative linear PDEs (11) is ensured, the rules of calculus that are displayed at the beginning of Section 2 are efficient. Again, this is in contrast with Otto calculus which only yields heuristics about displacement interpolations.
As regards (3), recall that we have seen at Theorems 5.4 and 5.8 unified proofs of entropy-entropy production inequalities, available in the diffusion and random walk settings.
In order to provide a better understanding of the analogy between displacement and entropic interpolations, we start describing two thought experiments. Then, keeping the LSV strategy as a guideline, we raise a few open questions about the use of entropic interpolations in connection with curvature problems on Riemannian manifolds and graphs.
Thought experiments. Let us describe two thought experiments.
• SchrÃűdinger's hot gas experiment. The dynamical SchrÃűdinger problem (see (S dyn ) at Section 1) is a formalization of SchrÃűdinger's thought experiment which was introduced in [Sch31]. Ask a perfect gas of particles living in a Riemannian manifold X which are in contact with a thermal bath to start from a configuration profile µ 0 ∈ P(X ) at time t = 0 and to end up at the unlikely profile µ 1 ∈ P(X ) at time t = 1. For instance the gas may be constituted of undistinguishable mesoscopic particles (e.g. grains of pollen) with a random motion communicated by numerous shocks with the microscopic particles of the thermal bath (e.g. molecules of hot water). Large deviation considerations lead to the following conclusion. In the (thermodynamical) limit of infinitely many mesoscopic particles, the most likely trajectory of the whole mesoscopic particle system from µ 0 to µ 1 is the Rentropic interpolation [µ 0 , µ 1 ] R where R describes the random motion of the noninteracting (the gas is assumed to be perfect) mesoscopic particles. Typically, R describes a Brownian motion as in SchrÃűdinger's original papers [Sch31,Sch32].
For more detail see [Léo14,§6] for instance. • Cold gas experiment. The same thought experiment is also described in [Vil09, pp. 445-446] in a slightly different setting where it is called the lazy gas experiment. The only difference with SchrÃűdinger's thought experiment is that no thermal bath enters the game. The perfect gas is cold so that the paths of the particles are Riemannian geodesics in order to minimize the average kinetic action of the whole system: entropy minimization is replaced by quadratic optimal transport.
Connection between entropic and displacement interpolations. It is shown in [Léo12,Léoa] that displacement interpolations are limits of entropic interpolations when the frequency of the random elementary motions encrypted by R tends down to zero. With the above thought experiment in mind, this corresponds to a thermal bath whose density tends down to zero. Replacing the hot water by an increasingly rarefied gas, at the zero-temperature limit, the entropic interpolation Cold gas experiment. As the gas is (absolutely) cold, the particle sample paths are regular and deterministic. The trajectory of the mass distribution of the cold gas is the displacement interpolation [µ 0 , µ 1 ] disp which is the time marginal flow of a probability measure P on Ω concentrated on geodesics: where π ∈ P(X 2 ) is an optimal transport plan between µ 0 and µ 1 , see [Léo14]. Therefore, where for simplicity it is assumed that there is a unique minimizing geodesic γ xy between x and y (otherwise, one is allowed to replace δ γ xy with any probability concentrated on the set of all minimizing geodesics from x to y).
In a positively curved manifold, several geodesics starting from the same point have a tendency to approach each other. Therefore, it is necessary that the gas initially spreads out to thwart its tendency to concentrate. Otherwise, it would not be possible to reach a largely spread target distribution. The left side of Figure 1 below (taken from [Vil09]) depicts this phenomenon. This is also the case of the right side as one can be convinced of by reversing time. The central part is obtained by interpolating between the initial and final segments of the geodesics. On the other hand, in a negatively curved manifold, several geodesics starting from the same point have a tendency to depart form each other. Therefore, it is necessary that the gas initially concentrates to thwart its tendency to spread out. Otherwise, it would not be possible to reach a condensed target distribution. SchrÃűdinger's hot gas experiment. As the gas is hot, the particle sample paths are irregular and random. The trajectory of the mass distribution of the hot gas is the entropic interpolation [µ 0 , µ 1 ] R . An R-entropic interpolation is the time marginal flow of a probability measure P on Ω which is a mixture of bridges of R: where π ∈ P(X 2 ) is the unique minimizer of η → H(η|R 01 ) among all η ∈ P(X 2 ) with prescribed marginals µ 0 and µ 1 , see [Léo14]. Therefore, µ t = P t = X 2 R xy t π(dxdy) ∈ P(X ), 0 ≤ t ≤ 1.
Comparing (59) and (61), one sees that the deterministic evolution δ γ xy is replaced by the random evolution R xy .
x y γ xy R xy t = 0 t = 1 Figure 3. Hot gas experiment with µ 0 = δ x and µ 1 = δ y The grey leaf in this figure is a symbol for the (say) 95%-support of the bridge R xy . It spreads around the geodesic γ xy and in general the support of R xy t for each intermediate time 0 < t < 1 is the whole space X . At a lower temperature, R xy is closer to δ γ xy : x y γ xy R xy t = 0 t = 1 Figure 4. Hot gas experiment with µ 0 = δ x and µ 1 = δ y . At a lower temperature And eventually, at zero temperature R xy must be replaced by its determinestic limit δ γ xy . The superposition of Figures 2 and 3 describes the hot gas experiment in a negative curvature manifold: Open questions. As far as one is interested in the curvature of the metric measure space (X , d, m), rather than in the curvature of some Markov generator − → L such as in Section 5, it seems natural to restrict our attention to a reversible reference path measure R. For instance when looking at a Riemannian manifold X considered as a metric measure space (X , d, m) with the Riemannian metric d and the weighted measure m = e −V vol, the efficient choice as regards LSV theory is to take R as the reversible Markov measure with generator L = (∆ − ∇V · ∇)/2 (or any positive multiple) and initial measure m.
A representative result in the case where the reference measure m is the volume measure (V = 0) is that relative entropy is convex along any displacement interpolation in a Riemannian manifold with a nonnegative Ricci curvature. Based on McCann's seminal work [McC94,McC97] and Otto's heuristic calculus, Otto and Villani have conjectured this result in [OV00]. This was proved by Cordero-Erausquin, McCann and Schmucken-schlÃďger in [CEMS01]. The converse is also true, as was shown later by Sturm and von Renesse [Sv05].
In the same spirit, an immediate consequence of Theorem 3.5 is the following Corollary 6.1. Let R o be the reversible Brownian motion on a connected Riemannian manifold X without boundary. Then, along any entropic interpolation [µ 0 , µ 1 ] R o associated with R o and f 0 , g 1 ∈ L 2 (vol) such that (8) holds, the entropy t ∈ [0, 1] → H(µ t |vol) ∈ R is a convex function.
Remark that in Theorem 3.5, X is assumed to be compact and f 0 , g 1 must be C 2 . This was assumed to ensure the regularity of the interpolation (Proposition 3.4). But in the present setting this regularity follows from the regularity of the heat kernel.
A definition of curvature should be justified by its usefulness in terms of rate of convergence to equilibrium, concentration of measure or isoperimetric behavior.

Appendix A. Basic definitions
This appendix section is part of the articles [Léob, LRZ]. We give it here for the reader's convenience.
Markov measures. We slightly extend the notion of Markov property to unbounded positive measures on Ω.
Definitions A.1. Any positive measure on Ω is called a path measure. Let Q be a path measure. (a) It is said to be a conditionable path measure if for any 0 ≤ t ≤ 1, Q t is a σ-finite measure on X . (b) It is said to be a Markov measure if it is conditionable and for all 0 ≤ t ≤ 1 and B ∈ σ(X [t,1] ), we have: Q(B | X [0,t] ) = Q(B | X t ). When Q has an infinite mass, the notion of conditional expectation must be handled with care. We refer to the following definition.
It is essential in this definition that the measure φ # Q is σ-finite on Y to be allowed to invoke Radon-Nikodym theorem when proving the existence of θ f . Inspecting the above definition of a Markov measure, one sees that it is necessary that Q [0,t] and Q t are σ-finite for all t ∈ [0, 1] for the corresponding conditional expectations to be defined. But this is warranted by the requirement that Q is conditionable, as shown by the following result. Lemma A.3. Let Q be a path measure and T ⊂ [0, 1]. For Q T to be σ-finite, it is enough that Q to is σ-finite for some t o ∈ T .
Proof. Let t o ∈ T be such that Q to is σ-finite with (X n ) n≥1 an increasing sequence of measurable subsets of X such that Q to (X n ) < ∞ and ∪ n≥1 X n = X . Then, Q T is also σ-finite since Q T (X to ∈ X n ) = Q to (X n ) for all n ≥ 1 and ∪ n≥1 [X T (Ω) ∩ {X to ∈ X n }] = X T (Ω).
Consequently, for any Markov measure Q and any T ⊂ [0, 1], the conditional expectation E Q (· | X T ) is well-defined and since Ω is a Polish space, there exists a regular conditional probability kernel Q(· | X T ) : Ω → P(Ω) such that for all f ∈ L 1 (Q), E Q (f | X T ) = Ω f dQ(· | X T ), see [Dud02, Thm. 10.2.2].