An entropic interpolation proof of the HWI inequality

Introduction

In a seminal paper , Otto and Villani obtained a powerful functional inequality relating the relative entropy $H(\cdot\,|\,\mathfrak{m})$ with respect to some reference measure $\mathfrak{m}\in\mathcal{P}_{2}({\rm X})$, the quadratic transport cost $W_{2}^{2}(\cdot,\mathfrak{m})$ and the Fisher information $I(\cdot\,|\,\mathfrak{m})$. This so-called HWI inequality roughly states that: $H\leq W\sqrt{I}-\kappa W^{2}/2,$ where the real parameter $\kappa$ is a curvature lower bound associated to $\mathfrak{m}$, see (1.2), (1.4) and Theorem 1.6 below for the exact statement and its well-known consequences in terms of Talagrand and logarithmic Sobolev inequalities.

The first part of Otto and Villani’s article is dedicated to a heuristic proof based on Otto calculus, see , where one formally equips the set of probability measures with the Riemannian-like distance $W_{2}$ and where McCann displacement interpolations are interpreted as geodesics. Since these interpolations suffer from a lack of regularity, the first and second order time derivatives along them are only formal. Consequently, although heuristics led to the right conjecture, the authors presented an alternative rigorous proof based on a significantly different approach.

In the present article, a new proof of the HWI inequality is proposed. The main idea is to replace the ‘irregular’ McCann interpolation $(\mu_{t})$ between two probability measures $\mu_{0}$ and $\mu_{1}$ by a family of ‘smooth’ curves of measures $(\mu_{t}^{\varepsilon})$, called ‘entropic interpolations’ (see Definition 2.3 below), where $\varepsilon>0$ is a small fluctuation parameter such that $\mu_{t}^{\varepsilon}$ converges to $\mu_{t}$ narrowly as $\varepsilon\downarrow 0$. Otto and Villani’s heuristics apply rigorously to $(\mu_{t}^{\varepsilon})$, so that it remains to let $\varepsilon$ tend down to zero to obtain the desired result.

The paper is structured as follows. Section 1 is dedicated to the statement of the HWI inequality. Basic material about entropic interpolations which is needed for the proof is gathered at Section 2. Finally the proof of the inequality is done at Section 3; its core is Lemma 3.11 which is the analogue of Otto and Villani’s heuristic approach. In Section 4 some comments about possible extensions and simplifications of our approach are collected.

Statement of the HWI inequality

Before stating the HWI inequality at Proposition 1.5 and Theorem 1.6 below, we need to make clear the framework we shall work within and introduce the quantities $H$, $W$ and $I$.

or $(M,{\sf d}_{g},\mathfrak{m})$, where $M$ is a smooth Riemannian manifold without boundary and with metric tensor $g$, ${\sf d}_{g}$ is the induced distance and $\mathfrak{m}$ is given by

Relative entropy

For any two probability measures $p$ and $r$ on a measurable space $Z$ the relative entropy of $p$ with respect to $r$ is defined by

where it is understood that this quantity is infinite when $p$ is not absolutely continuous with respect to $r$. In our case, $Z$ will be ${\rm X}$, ${\rm X}\times{\rm X}$ or $C(,{\rm X})$.

Quadratic transport cost

Fisher information

The Fisher information of $\mu\in\mathcal{P}({\rm X})$ with respect to $\mathfrak{m}$ is defined by

and $+\infty$ otherwise. Up to identify $\mu$ with its density, the Fisher information is lower semicontinuous with respect to the weak topology of $L^{1}(\mathfrak{m})$ (see for instance ).

With this premise, the statement of the HWI∗ inequality is

Let $({\rm X},{\sf d},\mathfrak{m})$ be as in Setting 1. Then for any $\mu_{0},\mu_{1}\in\mathcal{P}_{2}({\rm X})$ such that $H(\mu_{0}\,|\,\mathfrak{m})<\infty$,

In this case it is said that the reference measure $\mathfrak{m}$ satisfies the HWI inequality.

As already shown by Otto and Villani in , different choices of $\mu_{0}$ and $\mu_{1}$ in Proposition 1.5 entail three important consequences collected here below.

Let $({\rm X},{\sf d},\mathfrak{m})$ be as in Setting 1 with the further assumption that $\mathfrak{m}\in\mathcal{P}_{2}({\rm X})$. Then the following inequalities are satisfied:

Logarithmic Sobolev inequality: assume that $\kappa>0$, then

First of all, since $\mathfrak{m}\in\mathcal{P}_{2}({\rm X})$, it follows that $W_{2}(\nu,\mathfrak{m})$ is finite for all $\nu\in\mathcal{P}_{2}({\rm X})$.

The HWI inequality is obtained by choosing $\mu_{0}=\mathfrak{m},\mu_{1}=\nu$.

The Talagrand inequality is obtained by choosing $\mu_{0}=\nu,\mu_{1}=\mathfrak{m}$.

When $\kappa>0$, the logarithmic Sobolev inequality with $\nu\in\mathcal{P}_{2}({\rm X})$ follows by taking the supremum with respect to $W_{2}$ in the right-hand side of the HWI inequality (a). To extend this result to the case where $\nu\in\mathcal{P}({\rm X})$, a standard approximation argument (carried out for instance in Lemma 3.1 below) is sufficient.

When $\kappa>0$, any $\nu\in\mathcal{P}({\rm X})$ such that $H(\nu\,|\,\mathfrak{m})<\infty$ stands in $\mathcal{P}_{2}({\rm X})$. This follows from the variational representation of the relative entropy as

whence the claim. In particular, $\mathfrak{m}\in\mathcal{P}_{2}({\rm X})$.

Talagrand inequality (b) is irrelevant when $\kappa\leq 0$. When $\kappa>0,$ in view of previous remark it extends to all $\nu\in\mathcal{P}({\rm X})$ provided that one sets $W_{2}(\nu,\mathfrak{m})=\infty$ when $\nu$ does not belong to $\mathcal{P}_{2}({\rm X})$.

It follows from the logarithmic Sobolev inequality that when (1.2) or (1.4) holds with some $\kappa>0$, any $\nu\in\mathcal{P}({\rm X})$ such that $H(\nu\,|\,\mathfrak{m})=\infty$ satisfies $I(\nu\,|\,\mathfrak{m})=\infty$. Similarly, it follows from the HWI inequality that when (1.2) or (1.4) is only supposed to hold with $\kappa$ real, as soon as $\nu\in\mathcal{P}_{2}({\rm X})$, then $H(\nu\,|\,\mathfrak{m})=\infty$ implies that $I(\nu\,|\,\mathfrak{m})=\infty$.

Entropic interpolations

In this section we propose a short and self-contained presentation of entropic interpolations. The purpose is twofold: to provide the reader with those notions and results that will be frequently used later on and discuss their physical interpretation via Nelson’s dynamical view of diffusion processes. For sake of simplicity, the latter will be carried out in the more familiar Euclidean setting.

Let $X=(X_{t})_{0\leq t\leq 1}$, $X_{t}:\Omega\to{\rm X}$ with $\Omega:=C(,{\rm X})$ be the canonical process, defined by

For any path measure ${\sf Q}\in\mathcal{P}(\Omega)$ and each $0\leq t\leq 1$, we denote by ${\sf Q}_{t}:=(X_{t})_{\#}{\sf Q}\in\mathcal{P}({\rm X})$ the $t$-th marginal of ${\sf Q}$, that is the law of the position $X_{t}$ at time $t$ of the random path $X$ under ${\sf Q}$. Moreover, for any $0\leq s,t\leq 1$, we shall denote by ${\sf Q}_{st}$ the joint law of $X_{s}$ and $X_{t}$ under ${\sf Q}$, namely ${\sf Q}_{st}:=(X_{s},X_{t})_{\#}{\sf Q}$.

As reference path measure ${\sf R}\in\mathcal{P}(\Omega)$ we consider the law of the Markov diffusion process with generator

with initial law ${\sf R}_{0}=\mathfrak{m}$, where the potential $V$ appears at (1.1), (1.3), $\Delta$ is the Laplace-Beltrami operator on ${\rm X}$ and $\nabla$ the Levi-Civita connection associated to the metric $g$ (in Setting 1-(a) they are nothing but the standard Laplacian and gradient). It is well-known that ${\sf R}$ is a reversible Markov measure with reversing measure $\mathfrak{m}$. In particular it is stationary, that is ${\sf R}_{t}=\mathfrak{m}$ for all $0\leq t\leq 1$. For any $\varepsilon>0$ we denote by $X^{\varepsilon}$ the time-rescaled process defined by $X^{\varepsilon}_{t}:=X_{\varepsilon t}$, $0\leq t\leq 1$, and by ${\sf R}^{\varepsilon}:=(X^{\varepsilon})_{\#}{\sf R}$ the corresponding path measure. The parameter $\varepsilon$ is meant to tend to zero so that ${\sf R}^{\varepsilon}$ is a slowed down version of ${\sf R}$, whose generator is

For any $\varepsilon>0$, as a time rescaling of ${\sf R}$, ${\sf R}^{\varepsilon}$ is also $\mathfrak{m}$-reversible, so that in particular ${\sf R}^{\varepsilon}_{t}=\mathfrak{m}$ for all $t$.

The 1-parameter semigroup associated to $\mathcal{L}$ will be denoted by $(\mathcal{T}_{t})$ and, in a completely analogous way, $(\mathcal{T}^{\varepsilon}_{t})$ the one associated to $\mathcal{L}^{\varepsilon}$; notice that $\mathcal{T}^{\varepsilon}_{t}=\mathcal{T}_{\varepsilon t}$ for all $t\geq 0$. Within Setting 1 it is well-known (see for instance ) that there exists a unique heat kernel ${\sf r}_{t}(x,y)$ associated to $\mathcal{L}$ which is a smooth function on $(0,\infty)\times{\rm X}\times{\rm X}$. Therefore, the semigroup $(\mathcal{T}_{t})$ can be represented by

for all $f\in L^{\infty}(\mathfrak{m})$. Let us also recall that, in conjunction with (1.2) or (1.4), $(\mathcal{T}_{t})$ enjoys the Bakry-Émery contraction estimate

For its proof as well as for all the regularizing properties of $\mathcal{T}$ that will be used throughout the paper, we address the reader to .

defined for all $f,g\in C^{\infty}_{c}({\rm X})$, are naturally associated to $\mathcal{L}$. As it is not difficult to see, the drift $V$ in $\mathfrak{m}$ does not affect $\Gamma$, since $\Gamma(f,g)=\langle\nabla f,\nabla g\rangle$. It is worth mentioning that, with respect to the standard definition provided in , here $\Gamma$ and $\Gamma_{2}$ are not divided by 2, as the factor 1/2 already appears in $\mathcal{L}$, which thus corresponds to an SDE driven by a standard Brownian motion.

The Schrödinger problem

Let $\mu_{0},\mu_{1}\in\mathcal{P}({\rm X})$ be two probability measures: the Schrödinger problem associated with ${\sf R}^{\varepsilon},\mu_{0},\mu_{1}$ is defined by

and its value is called ‘entropic cost’. As a strictly convex minimization problem, it admits at most one solution.

The solution ${\sf P}^{\varepsilon}$ of (Sε), if it exists, is called the ${\sf R}^{\varepsilon}$-entropic bridge between $\mu_{0}$ and $\mu_{1}$. The ${\sf R}^{\varepsilon}$-entropic interpolation $(\mu^{\varepsilon}_{t})$ between $\mu_{0}$ and $\mu_{1}$ is defined as the time marginal flow of the solution ${\sf P}^{\varepsilon}$, namely

The name ‘entropic interpolation’ stems from the connection with displacement interpolation. Indeed, it is known from that

This limit is a consequence of a more general result asserting that (Sε) converges to the quadratic Monge-Kantorovich problem as $\varepsilon\downarrow 0$ in the sense of $\Gamma$-convergence, see . As shown in , if (Sε) admits the solution ${\sf P}^{\varepsilon}$, then there exist two non-negative measurable functions $f^{\varepsilon},g^{\varepsilon}:{\rm X}\to[0,\infty)$ such that ${\sf P}^{\varepsilon}=f^{\varepsilon}(X_{0})g^{\varepsilon}(X_{1})\,{\sf R}^{\varepsilon}$ and defining

usually known as ‘Schrödinger system’: indeed, if we interprete them as a nonlinear system where the unknowns are $f^{\varepsilon}$ and $g^{\varepsilon}$, then the (unique up to an obvious multiplicative rescaling) solution completely determines ${\sf P}^{\varepsilon}$ (see for instance ). As far as the convergence of entropic interpolations towards displacement ones is investigated, the following functions

are of special interest. We also set $\varphi^{\varepsilon}:=\varepsilon\log f^{\varepsilon}$ in $\mathop{\rm supp}\nolimits(\mu_{0})$ and $\psi^{\varepsilon}:=\varepsilon\log g^{\varepsilon}$ in $\mathop{\rm supp}\nolimits(\mu_{1})$. They are called Schrödinger potentials, in connection with Kantorovich ones.

Existence and regularity results

Let us now derive a criterion, in terms of the endpoint marginals $\mu_{0}$ and $\mu_{1}$, for the existence of regular functions $f^{\varepsilon},g^{\varepsilon}$ with well-defined Fisher information solving the Schrödinger system (2.6). As noticed in , the regularity (smoothness and integrability) of $\mu_{0}$ (resp. $\mu_{1}$) is inherited by $f^{\varepsilon}$ (resp. $g^{\varepsilon}$). In the next result we extend this property: if $I(\mu_{0}\,|\,\mathfrak{m})$ is finite, then so is $I(f^{\varepsilon}\mathfrak{m}\,|\,\mathfrak{m})$ and analogously with $\mu_{1},g^{\varepsilon}$.

Let $({\rm X},{\sf d},\mathfrak{m})$ be as in Setting 1, $\varepsilon>0$ and consider two probability measures $\mu_{0}=\rho_{0}\mathfrak{m},\mu_{1}=\rho_{1}\mathfrak{m}\in\mathcal{P}({\rm X})$ with compact supports. Then the following hold:

The Schrödinger problem (Sε) admits a solution with finite entropy if and only if $H(\mu_{0}\,|\,\mathfrak{m})$, $H(\mu_{1}\,|\,\mathfrak{m})<\infty$. This solution is unique and the ${\sf R}^{\varepsilon}$-entropic interpolation between $\mu_{0}$ and $\mu_{1}$ exists.

Suppose in addition that $\rho_{0},\rho_{1}\in L^{\infty}(\mathfrak{m})$.

Then $f^{\varepsilon},g^{\varepsilon}\in L^{\infty}(\mathfrak{m})$.

For any $0<t<1$ the functions $f^{\varepsilon}_{t},g^{\varepsilon}_{t},\rho^{\varepsilon}_{t}$ as well as $f^{\varepsilon}_{1}$ and $g^{\varepsilon}_{0}$ belong to $C^{\infty}({\rm X})\cap L^{\infty}(\mathfrak{m})$.

In addition to item (b), suppose that $I(\mu_{0}\,|\,\mathfrak{m})$ is finite (resp. $I(\mu_{1}\,|\,\mathfrak{m})$). Then so is $I(f^{\varepsilon}\mathfrak{m}\,|\,\mathfrak{m})$ (resp. $I(g^{\varepsilon}\mathfrak{m}\,|\,\mathfrak{m})$).

For (a) and (b-i) see [13, Proposition 2.1]. As regards (b-ii), the fact that $f^{\varepsilon}\in L^{\infty}(\mathfrak{m})$, ${\sf r}_{t}\in C^{\infty}({\rm X}\times{\rm X})$ and (2.1) imply $f_{t}^{\varepsilon}\in C^{\infty}({\rm X})$ for all $0<t\leq 1$, while the maximum principle ensures that $f_{t}^{\varepsilon}\in L^{\infty}(\mathfrak{m})$; the statements for $g^{\varepsilon}_{t}$ and $\rho_{t}^{\varepsilon}=f_{t}^{\varepsilon}g_{t}^{\varepsilon}$ follow by the same reason.

(b-iii). Notice that the first equation in the Schrödinger system (2.6) can be rewritten as

Since $\mathcal{T}^{\varepsilon}$ is positivity improving, $\mathcal{T}^{\varepsilon}_{1}g^{\varepsilon}$ is smooth and $\rho_{0}\in C^{k}({\rm X})$ with compact support, the conclusion follows. A similar argument holds for $g^{\varepsilon}$.

(c). Observe that the Schrödinger system (2.6) and $g_{0}^{\varepsilon}>0$ (as $\mathcal{T}^{\varepsilon}$ is positivity improving) force $f^{\varepsilon}$ to have the same support as $\rho_{0}$. Thus $f^{\varepsilon}$ has compact support and, as a consequence, $g_{0}^{\varepsilon}\geq c>0$ in $\mathop{\rm supp}\nolimits(f^{\varepsilon})$ for some $c$. This remark and the chain rule allow us to say that

so that it remains to prove the integrability of the right-hand side. The first term is integrable by assumption, the third one by the regularization properties of $\mathcal{T}^{\varepsilon}$, while for the second one notice that $I(\mu_{0}\,|\,\mathfrak{m})<\infty$ and $\rho_{0}\in L^{\infty}(\mathfrak{m})$ imply $|\nabla\rho_{0}|\in L^{2}(\mathfrak{m})$; plugging this information into

For this reason we formulate the following

The endpoint marginals $\mu_{0},\mu_{1}\in\mathcal{P}_{2}({\rm X})$ are such that $\mu_{0},\mu_{1}$ have compact supports, $H(\mu_{0}\,|\,\mathfrak{m}),H(\mu_{1}\,|\,\mathfrak{m}),I(\mu_{1}\,|\,\mathfrak{m})<\infty$ and their densities $\rho_{0},\rho_{1}$ belong to $C^{\infty}({\rm X})$.

A dynamical viewpoint

As concerns the evolution of entropic interpolations and Schrödinger potentials, let us first notice that under Assumptions 2.8, by the very definition (2.5), item (b-iii) of Proposition 2.7 and the fact that ${\sf r}_{t}(x,y)\in C^{\infty}((0,\infty)\times{\rm X}^{2})$ we deduce that $f_{t}^{\varepsilon}$ and $g_{t}^{\varepsilon}$ are smooth on $\times{\rm X}$ and solve

in the classical sense. Moreover, if we look at $(f_{t}^{\varepsilon}),(g_{t}^{\varepsilon})$ as curves parametrized by $t$ with values in $W^{1,2}({\rm X},\mathfrak{m})$, they belong to the set $AC(,W^{1,2}({\rm X},\mathfrak{m}))$ of all absolutely continuous functions from $$to$W^{1,2}({\rm X},\mathfrak{m})$. As a consequence the PDEs above hold also when$\partial_{t}f_{t}^{\varepsilon},\partial_{t}g_{t}^{\varepsilon}$are seen as strong$W^{1,2}$-limits.

Relying on that, it follows that the Schrödinger potentials $\varphi_{t}^{\varepsilon},\psi_{t}^{\varepsilon}$ are smooth on $(0,1]\times{\rm X}$ and $[0,1)\times{\rm X}$ and solve forward and backward Hamilton-Jacobi-Bellman equations respectively, i.e.

while for $(\rho_{t}^{\varepsilon})$ the continuity equation

is satisfied in $(0,1)\times{\rm X}$, where $\vartheta_{t}^{\varepsilon}:=(\psi_{t}^{\varepsilon}-\varphi_{t}^{\varepsilon})/2$ and ${\rm div}_{\mathfrak{m}}$ denotes the divergence with respect to $\mathfrak{m}$, i.e. the opposite of the adjoint of the differential in $W^{1,2}({\rm X},\mathfrak{m})$. This last PDE is strongly linked to the dynamical representation of the entropic cost $\inf\eqref{eq-h06}$, namely

shown in for Setting 1-(a) and in for rather general metric measure spaces including Setting 1-(b). By the very definition of $\vartheta_{t}^{\varepsilon}$ and since $\varepsilon\log\rho_{t}^{\varepsilon}=\varphi_{t}^{\varepsilon}+\psi_{t}^{\varepsilon}$, this implies

The regularity of Schrödinger potentials comes from the one of $f_{t}^{\varepsilon},g_{t}^{\varepsilon}$, the fact that $f_{t}^{\varepsilon},g_{t}^{\varepsilon}$ are everywhere positive and the logarithm is smooth on $(0,\infty)$. However, Schrödinger potentials are not integrable in general, as $f_{t}^{\varepsilon},g_{t}^{\varepsilon}$ can be arbitrarily close to 0. Thus we cannot study their behaviour as curves with values into some $L^{p}(\mathfrak{m})$ space. This also explains why the regularity of $\varphi_{t}^{\varepsilon}$ (resp. $\psi_{t}^{\varepsilon}$) does not extend up to $t=0$ (resp. $t=1$).

A physical interpretation

In the Euclidean framework of Setting 1-(a) it is possible to make a bridge between what is presented so far and Nelson’s formalism , thus providing a physical motivation for some results stated above and a further perspective on some objects.

For ${\sf R}^{\varepsilon}$, it is easily seen that

This allows to rewrite the continuity equation (2.11) as

where now ${\rm div}$ denotes the divergence with respect to $\mathcal{L}^{n}$ if in Setting 1-(a) or ${\rm Vol}$ if in Setting 1-(b). This is perfectly coherent with (2.11) since ${\rm div}_{\mathfrak{m}}(w)={\rm div}(w)-\frac{\varepsilon}{2}\nabla V\cdot w$ for any vector field $w$. Furthermore, (2.12) becomes

Proof of Proposition 1.5

We need to state some preliminary lemmas before completing the proof of Proposition 1.5 at page 3. Throughout the whole section we shall assume to work within $({\rm X},{\sf d},\mathfrak{m})$ as in Setting 1.

Let us start with an approximation result.

Let $\mu\in\mathcal{P}_{2}({\rm X})$ with $H(\mu\,|\,\mathfrak{m})<\infty$. Then:

there exists a sequence $(\mu_{n})\subset\mathcal{P}_{2}({\rm X})$ with $\mu_{n}=\rho_{n}\mathfrak{m}$ and $\rho_{n}\in C^{\infty}_{c}({\rm X})$ such that $W_{2}(\mu_{n},\nu)\to W_{2}(\mu,\nu)$ for all $\nu\in\mathcal{P}_{2}({\rm X})$ and $H(\mu_{n}\,|\,\mathfrak{m})\to H(\mu\,|\,\mathfrak{m})$ as $n\to\infty$;

if in addition $I(\mu\,|\,\mathfrak{m})<\infty$, then there exists a sequence $(\mu^{\prime}_{n})\subset\mathcal{P}_{2}({\rm X})$ with $\mu_{n}=\rho_{n}\mathfrak{m}$, $\rho_{n}\in C^{\infty}_{c}({\rm X})$ such that $W_{2}(\mu^{\prime}_{n},\nu)\to W_{2}(\mu,\nu)$ for all $\nu\in\mathcal{P}_{2}({\rm X})$, $H(\mu^{\prime}_{n}\,|\,\mathfrak{m})\to H(\mu\,|\,\mathfrak{m})$ and $I(\mu^{\prime}_{n}\,|\,\mathfrak{m})\to I(\mu\,|\,\mathfrak{m})$ as $n\to\infty$.

Let us write $\mu=\rho\,\mathfrak{m}$ and, as a first step, let us prove that both in (a) and (b) it is possible to find a sequence of measures with smooth densities converging to $\mu$ in the desired sense. This can be proved by defining for $\varepsilon>0$

which clearly have smooth densities by the regularizing properties of $(\mathcal{T}_{\varepsilon})$. The convergence of $W_{2}(\mu_{\varepsilon},\nu)$, $H(\mu_{\varepsilon}\,|\,\mathfrak{m})$ and $I(\mu_{\varepsilon}\,|\,\mathfrak{m})$ to $W_{2}(\mu,\nu)$, $H(\mu\,|\,\mathfrak{m})$ and $I(\mu\,|\,\mathfrak{m})$ respectively as $\varepsilon\downarrow 0$ is now a well-known fact in the theory of gradient flows (see for instance Theorem 2.4.15 and Remark 2.4.16 in in conjunction with the fact that the squared slope of the entropy is the Fisher information, as proved in ).

Thus, it is not restrictive to suppose that $\mu$ has smooth density. Under this new assumption, let us prove that we can find a sequence of measures with compact supports and smooth densities converging to $\mu$ in the desired sense. To this aim define

where $\alpha_{n}$ is the renormalization constant and $\chi_{n}$ is a smooth cut-off function with support in $B_{n+1}(x)$, for some $x\in{\rm X}$, $\chi_{n}(x)=1$ and Lipschitz constant controlled by $C/n$, where $C>1$ does not depend on $n$ (see e.g. for a proof of the existence of such cut-off functions). By dominated convergence it is not difficult to see that $W_{2}(\mu,\mu_{n})\to 0$ and thus $W_{2}(\mu_{n},\nu)\to W_{2}(\mu,\nu)$ for all $\nu\in\mathcal{P}_{2}({\rm X})$ as $n\to\infty$; for the same reason $H(\mu_{n}\,|\,\mathfrak{m})\to H(\mu\,|\,\mathfrak{m})$. If we also assume that $I(\mu\,|\,\mathfrak{m})<\infty$, then

Since the right-hand side is integrable and $\chi_{n}\to 1$ as $n\to\infty$, by dominated convergence we get $I(\mu_{n}\,|\,\mathfrak{m})\to I(\mu\,|\,\mathfrak{m})$.

Combining the two steps and using a diagonal argument, the conclusion follows. ∎

The following conservation result was pointed out in and in the case ${\rm X}$ is compact with different approaches; see also . Following , we extend the statement to the present framework.

Under Assumptions 2.8, for any $\varepsilon>0$ the function

is real-valued and constant. Thus we shall denote it by $Q^{\varepsilon}$.

As a first step, for all $0<t<1$ by algebraic manipulation we have

From integration by parts formula it is straightforward to see that the right-hand side vanishes, whence the conclusion. ∎

Motivated by (2.12), let us investigate separately the convergence of current and osmotic velocities as $\varepsilon\downarrow 0$.

Under Assumptions 2.8, for any $\varepsilon>0$ we have

Let us first notice that combining (2.12) and (2.4) we get

Since the continuity equation (2.11) is satisfied by $\mu_{t}^{\varepsilon}$, the Benamou-Brenier formula holds for any $\varepsilon>0$, that is

and together with the above limit, this leads us to the first identities both in (3.4) and (3.5). From them we immediately deduce that

whence also the second identity in (3.5). Finally observe that

As already noticed at Remark 2.14, although it is smooth on the open interval $(0,1)$, the density $\rho^{\varepsilon}_{t}$ might be arbitrarily close to zero. Consequently, the Schrödinger potentials might not be integrable enough, and the Fisher information $I(\mu^{\varepsilon}_{t}\,|\,\mathfrak{m})$ might behave badly around $t=0$ and $t=1$. Next lemma provides related regularity and growth controls. Its proof is strongly inspired by , , and ; we thus address the reader to these articles for more details.

Under Assumptions 2.8, let $\delta>0$ and set, for all $0\leq t\leq 1$, $\rho_{t}^{\varepsilon,\delta}:=(f_{t}^{\varepsilon}+\delta)(g_{t}^{\varepsilon}+\delta)$ as well as

Then all the functions so defined belong to $C^{\infty}(\times{\rm X})\cap W^{1,2}({\rm X},\mathfrak{m})$ and, as curves parametrized by $t$ with values in $W^{1,2}({\rm X},\mathfrak{m})$, to $AC(,W^{1,2}({\rm X},\mathfrak{m}))$. The time derivatives of $\varphi_{t}^{\varepsilon,\delta},\psi_{t}^{\varepsilon,\delta},\rho_{t}^{\varepsilon,\delta}$ are given by

where $\partial_{t}\varphi_{t}^{\varepsilon,\delta}$, $\partial_{t}\psi_{t}^{\varepsilon,\delta}$, $\partial_{t}\rho_{t}^{\varepsilon,\delta}$ have to be understood both in the classical sense and as strong $W^{1,2}$-limits.

As already explained in Section 2, under Assumptions 2.8 $f_{t}^{\varepsilon},g_{t}^{\varepsilon}\in C^{\infty}(,{\rm X})\cap W^{1,2}({\rm X},\mathfrak{m})$ and $(f_{t}^{\varepsilon}),(g_{t}^{\varepsilon})\in AC(,W^{1,2}({\rm X},\mathfrak{m}))$. Therefore the regularity and integrability properties of $\varphi_{t}^{\varepsilon,\delta},\psi_{t}^{\varepsilon,\delta},\vartheta_{t}^{\varepsilon,\delta},\rho_{t}^{\varepsilon,\delta}$ are a straightforward consequence of the chain rule and of the fact that the logarithm is smooth with bounded derivatives on $[\delta,\infty)$. Also the PDEs solved by $\varphi_{t}^{\varepsilon,\delta},\psi_{t}^{\varepsilon,\delta},\rho_{t}^{\varepsilon,\delta}$ are easily deduced, when interpreted in the classical sense, as they follow from (2.10) and (2.11). In order to deduce the same identities with $\partial_{t}\varphi_{t}^{\varepsilon,\delta}$, $\partial_{t}\psi_{t}^{\varepsilon,\delta}$, $\partial_{t}\rho_{t}^{\varepsilon,\delta}$ seen as strong $W^{1,2}$-limits, notice that by the maximum principle

whence $(\varphi_{t}^{\varepsilon,\delta})\in L^{\infty}((0,\infty),L^{\infty}(\mathfrak{m}))$. Moreover, the smoothness of the logarithm, the chain and Leibniz rules entail that

whence $(|\nabla\varphi_{t}^{\varepsilon,\delta}|^{2}),(\Delta\varphi_{t}^{\varepsilon,\delta})\in L^{\infty}((0,1),W^{1,2}({\rm X},\mathfrak{m}))$; analogous estimates hold for $\nabla|\nabla\varphi_{t}^{\varepsilon,\delta}|^{2}$ and $\nabla\Delta\varphi_{t}^{\varepsilon,\delta}$. These bounds together with the fact that the PDEs in (3.8) hold in the classical sense imply, by a dominated convergence argument, that (3.8) are satisfied also as strong $W^{1,2}$-limits.

Relying on that, (3.9a) and (3.9b) follow by the computations carried out in . Indeed, the validity of (3.8) as strong $W^{1,2}$-limits on $$ensures that$t\mapsto u(\rho_{t}^{\varepsilon,\delta})$and$t\mapsto\langle\nabla\rho_{t}^{\varepsilon,\delta},\nabla\vartheta_{t}^{\varepsilon,\delta}\rangle$belong to$AC(,L^{2}(\mathfrak{m}))$ and thus we can pass the time derivatives under the integral sign, i.e.

Then the Hamilton-Jacobi-Bellman equations for the Schrödinger potentials and the continuity equation for the entropic interpolation together with $\varepsilon\log\rho_{t}^{\varepsilon}=\varphi_{t}^{\varepsilon}+\psi_{t}^{\varepsilon}$, here replaced by $\varepsilon\log\rho_{t}^{\varepsilon,\delta}=\varphi_{t}^{\varepsilon,\delta}+\psi_{t}^{\varepsilon,\delta}$, are the only tools needed to deduce (3.9a) and (3.9b). Finally, the fact that:

$(\rho_{t}^{\varepsilon,\delta}),(\vartheta_{t}^{\varepsilon,\delta})\in AC(,W^{1,2}({\rm X},\mathfrak{m}))$;

$(\rho_{t}^{\varepsilon,\delta})\in AC(,W^{1,2}({\rm X},\mathfrak{m}))$, $(|\nabla\vartheta_{t}^{\varepsilon,\delta}|^{2}),(\Delta\vartheta_{t}^{\varepsilon,\delta}),(|\nabla\log\rho_{t}^{\varepsilon,\delta}|^{2}),(\Delta\log\rho_{t}^{\varepsilon,\delta})$ belong to $L^{\infty}((0,1),W^{1,2}({\rm X},\mathfrak{m}))$ and $\vartheta_{t}^{\varepsilon,\delta},\log\rho_{t}^{\varepsilon,\delta}\in C^{\infty}(\times{\rm X})$;

imply the continuity on $$ of the right-hand sides of (3.9a) and (3.9b) respectively. ∎

With this results at disposal we can prove our main lemma: a rigorous ‘entropic’ analogue of Otto-Villani’s heuristic argument.

Under Assumptions 2.8, for any $\varepsilon>0$ it holds

The proof of the lemma is based on the standard calculus identity

consequence of the lower Ricci bounds (1.2) and (1.4) rewritten in the form of the Bochner-Lichnerowicz-Weitzenböck formula, we obtain

where $\mu_{t}^{\varepsilon,\delta}:=\rho_{t}^{\varepsilon,\delta}\mathfrak{m}$. It is now sufficient to pass to the limit as $\delta\downarrow 0$. By dominated convergence (recall that $f^{\varepsilon},g^{\varepsilon}\in L^{\infty}(\mathfrak{m})$) it is easy to see that the left-hand side converges to $H(\mu_{1}\,|\,\mathfrak{m})-H(\mu_{0}\,|\,\mathfrak{m})$. In order to apply the dominated convergence theorem also to the first term on the right-hand side, notice that

Thus, to control $\langle\nabla\vartheta_{1}^{\varepsilon,\delta},\nabla\rho_{1}^{\varepsilon,\delta}\rangle$ let us first observe that

and remark that the right-hand side is integrable by Proposition 2.7-(c); secondly

and in this case the right-hand side is integrable as $g^{\varepsilon}$ has compact support and $f_{1}^{\varepsilon}$ is bounded away from 0 therein; finally,

as $g^{\varepsilon}+\delta\geq\delta$ and the same strategy applies to all the remaining terms that we obtain developing $\langle\nabla\vartheta_{1}^{\varepsilon,\delta},\nabla\rho_{1}^{\varepsilon,\delta}\rangle$. Therefore, the first term on the right-hand side of (3.14) converges to the first term on the right-hand side of (3.12). As regards the other two summands, by the very definition of $\vartheta_{t}^{\varepsilon,\delta}$ and since

the conclusion will follow if we are able to prove that

To this aim, notice that $\rho_{t}^{\varepsilon,\delta}=\rho_{t}^{\varepsilon}+\delta f_{t}^{\varepsilon}+\delta g_{t}^{\varepsilon}+\delta^{2}$, whence using either $f_{t}^{\varepsilon}+\delta\geq f_{t}^{\varepsilon}$ or $f_{t}^{\varepsilon}+\delta\geq\delta$ it is easy to infer that

All the right-hand sides above are integrable on $\times{\rm X}$ (either by (2.13) or by the Bakry-Émery contraction estimate (2.2) together with $f^{\varepsilon}\in C^{\infty}_{c}({\rm X})$), thus by dominated convergence (3.15a) follows. An analogous argument holds for (3.15b), whence the conclusion. ∎

Completion of the proof of Proposition 1.5

We are now ready to complete the proof of the HWI* inequality.

First of all, by Proposition 2.7 and Lemma 3.1, it is sufficient to prove the result for any $\mu_{0},\mu_{1}$ satisfying Assumptions 2.8. This allows us to invoke Lemma 3.11. Secondly, observe that by the Cauchy-Schwarz inequality

so that if we plug this information into (3.12) we obtain

Let us now pass to the limit as $\varepsilon\downarrow 0$: by Lemma 3.2, at $t=1$ we have

so that (3.6) together with $I(\mu_{1}\,|\,\mathfrak{m})<\infty$ yields

By Lemma 3.3 we can pass to the limit as $\varepsilon\downarrow 0$ also in the remaining terms on the right-hand side, thus concluding. ∎

Let us mention that another approach to prove the HWI inequality via the Schrödinger problem is pointed out in a recent work where an entropic counterpart of the HWI inequality is formally obtained by differentiating the convexity estimate of the entropy along the entropic interpolations introduced in [6, Thm. 1.4].

Final remarks and comments

From a heuristic point of view, the expression of the constant quantity $Q^{\varepsilon}$ can be deduced by standard arguments in Lagrangian and Hamiltonian formalism. Indeed, motivated by (2.12) let us consider the action functional

By means of Legendre’s transform, the corresponding Hamiltonian is given by

and, at least formally, $\mathscr{H}$ is constant along the critical points of $\mathscr{A}$. Since $\mu_{0}$ and $\mu_{1}$ are prescribed, the Euler-Lagrange equation for (4.1) reads as

and, as it is not difficult to see (e.g. following the computations carried out in which fit to Setting 1), these PDEs are satisfied along $(\rho_{t}^{\varepsilon},\vartheta_{t}^{\varepsilon})$. Finally, as in the Hamiltonian $p$ represents a momentum density, it is natural to set $p_{t}:=\nu_{t}v_{t}$. From these considerations, the guess on the existence of the conserved quantity of Lemma 3.2 and on its expression follows. This point of view is particularly investigated in the recent paper , to which we refer for more details.

The compact case

As already mentioned in Remark 2.14, in general Schrödinger potentials fail to be smooth in $t=0$ or $t=1$ and are not even integrable for any $0<t<1$, because $f_{t}^{\varepsilon},g_{t}^{\varepsilon}$ can be arbitrarily close to 0. This problem can be overcome if $f^{\varepsilon},g^{\varepsilon}\geq c>0$ for some constant $c$: then $f_{t}^{\varepsilon},g_{t}^{\varepsilon}\geq c$ as well by the maximum principle and $\varphi_{t}^{\varepsilon},\psi_{t}^{\varepsilon}\geq\varepsilon\log c$ from their very definition. However, even if we assume $\mu_{0},\mu_{1}$ to have densities bounded away from 0 (thus forcing $\mathfrak{m}$ to belong to $\mathcal{P}_{2}({\rm X})$, a condition which does not follow from our Setting 1, unless $\kappa>0$, cf. Remark 1.7-(a)), it is not known yet whether this implies an analogous lower bound on $f^{\varepsilon}$ and $g^{\varepsilon}$. This is the motivation behind the definitions provided in Lemma 3.7.

On the contrary, if ${\rm X}$ is assumed to be compact, then it is not restrictive to assume $\mu_{0},\mu_{1}\geq c\mathfrak{m}$ for some $c>0$: indeed any $\mu\in\mathcal{P}({\rm X})$ can be approximated by

(where $\alpha_{n}$ is the renormalization constant) in $W_{2}$, entropy and Fisher information in the sense of Lemma 3.1. Secondly, ${\sf r}_{\varepsilon}$ is bounded from above since ${\sf r}_{\varepsilon}\in C^{\infty}({\rm X})$. These two facts imply that $f^{\varepsilon},g^{\varepsilon}\geq c^{\prime}>0$, as shown in . As a consequence the proof of the HWI inequality is easier and the parallelism with the heuristic proof of Otto-Villani is stronger, since no ‘$\delta$-argument’ in Lemma 3.7 and Lemma 3.11 is needed; in particular, Lemma 3.7 is already known to hold by and thus no proof is required.

𝖱𝖢𝖣𝖱𝖢𝖣{\sf RCD} spaces

All the results presented in this paper, with particular mention of the key Lemma 3.11 and Proposition 1.5, are also true in a more general framework than the one of Setting 1-(b), namely in ${\sf RCD}^{*}(K,N)$ spaces (introduced in ). If ${\rm X}$ is assumed to be compact, then this has been shown by the last author in his PhD thesis . When ${\rm X}$ is not compact, the most important steps in our entropic approach still hold. Namely:

all the regularity and integrability results concerning Schrödinger potentials and entropic interpolations mentioned in Section 2 as well as the dynamic representation of the entropic cost;

the regularizing and contraction properties of $(\mathcal{T}_{t})$;

the existence of ‘good’ cut-off functions;

the Benamou-Brenier formula and the Bochner-Lichnerowicz-Weitzenböck inequality.

The reader is addressed to , for the first point and to for all the others.

Acknowledgements. This research was supported by the French ANR-17-CE40-0030 EFI project and the LABEX MILYON ANR-10-LABX-0070.

References