On the variational interpretation of local logarithmic Sobolev inequalities

The celebrated Otto calculus has established itself as a powerful tool for proving quantitative energy dissipation estimates and provides with an elegant geometric interpretation of certain functional inequalities such as the Logarithmic Sobolev inequality. However, the \emph{local} versions of such inequalities, which can be proven by means of Bakry-Emery-Ledoux $\Gamma$-calculus, has not yet been given an interpretation in terms of this Riemannian formalism. In this short note we close this gap by explaining how Otto calculus applied to the Schr{\"o}dinger problem yields a variations interpretation of the local logarithmic Sobolev inequalities, that could possibly unlock novel class of local inequalities.


Curvature dimension conditions
We are working on a (N, g), a smooth, connected and compact Riemannian manifold without boundary.The heat equation, ∂ t u = ∆u, starting from f is denoted (t, x) → P t f (x), where P t f (x) = f (y)p x t (y)dy, x ∈ N, and p x t (y) is the density of the heat kernel with respect to the Riemannian measure dy.∆ = ∇ • ∇ is the Laplace-Beltrami operator (∇• is the divergence operator whereas ∇ is the usual gradient operator).(P t ) t 0 is a reversible Markov semigroup in the sense of [BGL14].Next, we introduce the carré du champ operator Γ and its iterated form Γ 2 , they are defined by In particular, Böchner's formula yields the equivalent expression of Γ 2 in terms of Ricci's curvature: Following the seminal paper [BE85], we introduce for ρ ∈ R, n ∈ [1, ∞] the curvature-dimension condition CD(ρ, n) by asking that holds for any smooth function f .In view of Böchner's formula, the CD(ρ, n) condition is equivalent to the fact that Ric g ρ and d ≤ n where d is the dimension of N .We also need to introduce the dual semigroup (P * t ) t 0 acting on absolutely continuous measures µ ∈ P(N ) as follows P * t (µ) = P t dµ dx dx. (1) For the sake of clarity, we limit the discussion in the present note to the Laplace-Beltrami operator in a compact Riemannian Manifold instead a general diffusion Markov operator.However, all of the results we are going to present, can be formulated in such general context following the paradigm of [BGL14].

Bakry-Émery-Ledoux calculus and local logarithmic Sobolev inequalities
Bakry-Émery-Ledoux calculus, known as Γ calculus, has been used as a powerful tool to analyse the quantitative behaviour of the heat semigroup (P t ) t 0 in various ways.Among its main achievements are local logarithmic Sobolev inequalities, proven under CD(ρ, ∞) and the CD(0, n) conditions * .
1. Local inequalities under CD(ρ, ∞), cf.[Bak97].Under the CD(ρ, ∞) condition the following inequalities are valid for all positive functions g and T > 0: • The mild gradient commutation estimate (2) • The local logarithmic Sobolev inequality • The reverse local logarithmic Sobolev inequality P T (g log g) − P T g log P T g e 2ρT − 1 2ρ 2. Local inequality under the CD(0, n) condition, cf.[BL06].Under the CD(ρ, ∞) condition the following inequalities are valid for all positive functions g and T > 0: • The local logarithmic Sobolev inequality and in particular ∆P T g P • The reverse local logarithmic Sobolev inequality which yields as a by product the celebrated Li-Yau inequality [LY86], [BL06].
Remark 1.1 • The main interest of such inequalities is that there are local, in the sense that they give estimates on the heat kernel itself instead of the entropy of the heat kernel.Moreover, in contrast with their integrated versions, these inequalities imply back the corresponding curvature-dimension conditions, i.e. inequalities (2)-(4) imply CD(ρ, ∞) and (5),(7) imply CD(0, n).
• Many others inequalities have been proved using the same kind of method.Unfortunately, it is actually not possible to reach it with Otto Calculus and the Schrödinger problem.Let mention for instance the isoperimetric inequality under the CD(ρ, ∞) condition proved in [BL96], the local hypercontractivity inequality from [BBG12] etc.

Otto calculus
The term Otto calculus, introduced by C. Villani in his book [Vil09], refers to a Riemannian formalism on the space of probability measures M = P(N ) that allows, among other things, to interpret the Wasserstein distance as a geodesic distance and the heat equation as the gradient flow of the entropy.Developed in the seminal papers [JKO98,Ott01,OV00], it has led to major advances in the quantifying trend to equilibrium of possibly non linear PDEs, to the discovery of new functional inequalities as well as to a new geometric interpretation of classical ones.We briefly present here the main ideas that lead to the definition of the Riemannian metric known as Otto metric.Our heuristic presentation is based on [GLR20], to which we refer for more details (see also [Gen21] for an informal presentation in French).However, given its importance, there are many other references that provide with an introduction to Otto calculus, see e.g. the monograph [Gig12] for rigorous constructions, [AG13] for a user-friendly presentation and the book [AGS08, Chap.1].We begin by introducing the two main functionals of interest for this work: the Wasserstein distance and the relative entropy.The first one is defined for all µ, ν ∈ P 2 (N ) by where the infimum runs over all π ∈ P(N × N ) with marginals µ and ν and P 2 (N ) is the set of probability measures with finite second moment (since in our case N is compact, P 2 (N ) = P(N )).For any probability measure µ ∈ P(N ) the entropy with respect to the Riemannian measure is given by We recall that a path [0, 1] ∋ t → µ t ∈ P 2 (N ) is absolutely continuous if there exists a non negative function In that case, one can define Moreover, for any absolutely continuous path (µ t ) t 0 , there exists a unique vector field (t, x) and we call μt the velocity of the path (µ t ) t∈[0,1] at time t.Finally, the vector field V t is a weak solution of the continuity equation To give an example, in the case of the heat equation ∂ t ν t = ∆ν t , the velocity of the path (ν t The tangent space at µ ∈ P 2 (N ) is identified with The Riemannian metric on T µ P 2 (N ) is then defined via the scalar product L 2 (µ), ∇ϕ, ∇ψ µ = Γ(ϕ, ψ)dµ, ∇ϕ, ∇ψ ∈ T µ P 2 (N ).
Such metric is often referred to the Otto metric and it can be seen that the geodesics associated to the Otto metric are the displacement interpolations of optimal transport.Using this, a straightforward computation implies that the gradient of the entropy F at µ is given by Accordingly, we can rewrite the Fisher information functional I as where Γ(F ) can be interpreted as the carré du champ operator applied to the functional F .In light of (12), we can now view the semigroup (P * t ) t 0 as the gradient flow of the function F , that is to say νt = −grad νt F .Now, we turn our attention to the second order calculus introducing covariant derivatives and Hessians.A remarkable fact is that the Hessian of the entropy F can be expressed in terms of the Γ 2 operator.In fact, we have (see for instance [OV00] and [GLR20, Sec 3.3]), From the work of [EKS15], the curvature-dimension condition CD(ρ, n) (ρ ∈ R, n > 0) is equivalent to the differential inequality As in a finite dimensional Riemannian manifold, the acceleration of a curve is defined as the covariant derivative of the velocity field along the curve itself.Recalling the definition of velocity μt we gave at (10), it turns out that the acceleration, which we denote μt is given by where μt = ∇ϕ t in the above equation.

Some simple inequalities on a toy model
In this section we show a series of inequalities inspired by a toy model proposed in [GLR20].This model, in a very simple way, captures some geometric features of the Schrödinger problem.Such inequalities will translate effortlessly to the true Schrödinger problem by means of Otto calculus, at least formally, and in this latter form they will be seen to be equivalent to local logarithmic inequalities, thus delivering the main message of this note.We shall do so in Section 3.
To build the toy model we consider a function F : R d −→ R be a twice differentiable function with d > 0. We note F ′ (resp.F ′′ ) the gradient (resp.the Hessian) of F .For every T > 0 and x, y ∈ R d , the toy model is the following optimization problem where the infimum taken over all smooth paths from [0, T ] to R d such that ω 0 = x and ω T = y.A standard variational argument shows that any minimizer X T t t∈[0,T ] of (16) satisfies Newton's system and is called an F -interpolation between x and y.From (17) we also deduce that is conserved along optimal curves, i.e. it is a constant function of t.E T plays the role of the conserved total (kinetic + potential) energy of a physical system.Mimicking (14), we say that F is (ρ, n)-convex if We proceed to show how the announced inequalities hold under either in the (ρ, ∞) or the (0, n) setting.In both cases, they will be consequences of convexity estimates for the function where (X T t ) t∈[0,T ] is an F -interpolation.Note that λ(T ) = C T (x, y) + 2(F (y) − F (x)).Note that λ(t) is a measure of how much the F -interpolation deviates from being a gradient flow.We have therefore established the inequalities 2. Estimates under the (0, n)-convexity with n > 0, cf.[CCG21].In this case we consider the map Φ(t) = λ(t) − tE T (x, y) and observe that (0, n) and Newton's law combined yield which is equivalent to concavity of the map [0, T ] ∋ t → exp − Φ(t) 2n .then, using the basic convexity estimates In particular,

The Schrödinger problem
We now move on to analyse the Schrödinger problem which we obtain from the toy model discussed above by replacing the underlying space is P 2 (N ) and equipping it with the Otto metric, see [GT20,GLR20].We remark that this is not the original formulation of the Schrödinger problem which, by means of Sanov's theorem and large deviations theory is usually cast as an entropy minimization problem on path space, see the survey [Leo14].
Let us now recall some fundamental properties of entropic interpolations.For a detailed account, we refer to [Leo14].
• For any µ, ν ∈ P(N ), absolutely continuous with a positive C ∞ density with respect to dx, then there exists f, g ∈ C ∞ (N, (0, ∞)) such that minimizers of (19) (µ T t ) t∈[0,T ] are given by, Moreover, the couple (f, g) is unique up to a multiplicative constant and the velocity field of the entropic interpolation is given by • Entropic interpolations are solution to a second order equation, akin to Newton's law (17).More precisely, it has been proven in [Con19, Theorem 1.2] (see also [GLR20,Sec. 3.3] and let us notice that the acceleration of the entropic interpolation has been computed previously in [GT21]) that the entropic interpolation where (μ T t ) t∈[0,T ] is the acceleration defined in (15), that is completely analogous to Newton's law (17).Moreover, the quantity It remains to write the inequalities derived for the toy model in terms of entropic interpolations.By replacing the function λ(t) used in section 2 with the function where (µ T s ) t∈[0,T ] is an entropic interpolation.We obtain, arguing exactly as before but working on P 2 (N ) and using the Riemannian formalism associated with the Otto metric the following estimates 1. Estimates under CD(ρ, ∞) with ρ ∈ R, cf.[Con19,GLR20].In this case we have that if (µ T t ) t∈[0,T ] is the entropic interpolation between µ and ν: 2. Estimates under the (0, n)-convexity with n > 0, cf.[CCG21].In this case we have that if (µ T t ) t∈[0,T ] is the entropic interpolation between µ and ν: In particular,

Local Logarithmic Sobolev inequalities via Schrödinger problem and Otto calculus
The main contribution of this note is to show how the Bakry-Émery-Ledoux estimates (2) to (8) are in fact equivalent to the inequalities derives in Section 3 along entropic interpolations.The key to do this is the product formula (20).
Then, by reversibility of the semigroup (P t ) t 0 , inequality (24) can be written as But, for any positive functions f, g, the path is an interpolation entropic, then the previous inequality (31) is valid for any functions f, g > 0. If we now let f dy → δ x weakly, we get Γ(P T g) P T g (y) ≤ e −2ρT P T Γ(g) g (y), which is precisely (2).
So, at the end, By using the path defined in (32), one can conclude that inequality (25) can be written as for any functions f, g > 0. Again, if f dx → δ y (y ∈ N ) weakly, the previous inequality becomes which is exactly the local logarithmic Sobolev inequality (3).Arguing similarly, inequality (26) is seen to be equivalent to the local reverse logarithmic Sobolev inequality (4).
Local logarithmic Sobolev inequalities (5)-(6) under CD(0, n) and the Li-Yau inequality (8).To understand the last inequalities, is it enough to compute the conserved quantity.We have, Again, if we let f dx → δ y (y ∈ N ), we get inequality (5).A similar argument can be used for the equivalence between (28) and the reverse local inequality (7) and to show the equivalence of (29) and the Li-Yau inequality (8).
Remark 4.1 When µ = δ y for some y ∈ N and ν ∈ P (N ), the entropic interpolation is explicit and given by the following path [0, T ] ∋ t → µ T t = p y t P T −t dν dx It is interesting to notice that it is in fact, the path used in the computations of this note.

Conclusion
We have demonstrated how local logartihmic Sobolev inequalities (2)-( 7) admit a nice geometrical interpretation via inequalities (24)-( 28) by means of Otto calculus and the Schrödinger problem.Such interpretation was first given for the integrated version of the logarithmic Sobolev inequality in [JKO98] and our results are the first, to the best of our knowledge to cover the local versions.We hope that our considerations will be of help to understand local inequalities for general functionals satisfying the curvature dimension conditions beyond, thus going beyond the case of the Boltzmann entropy.
1. Estimates under (ρ, ∞)-convexity with ρ ∈ R, cf.[Con19,GLR20].A straightforward calculation using (ρ, ∞)-convexity and Newton's law yields that λ verifies the differential inequality Using the path of probability measures (32), for any functions f, g > 0, we have exp 2 n P T gf dx [P T (g log g) − P T g log P T g]f dx − T ∆P T gf dx t g P t f − Γ(log P T −t gP t f ) P T −t gP t f dx = − 4 Γ(log P T −t g, log P t f )P T −t gP t f dx = −4 Γ(P T −t g, P t f ) dx = 4 ∆P T gf dx.(34) T (g log g) − P T g log P T g]f dx − 4T ∆P T gf dx T gf dx .