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\TopicFR{Analyse fonctionnelle}
\TopicEN{Functional analysis}
\TopicFR{Probabilités}
\TopicEN{Probability theory}

\addbibresource{CRMATH_Courtade_20240576.bib}


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\title{Stochastic proof of the  sharp  symmetrized Talagrand inequality}

\alttitle{Une preuve stochastique de l'inégalité de Talagrand symétrisée optimale}

\author{\firstname{Thomas} \middlename{A.} \lastname{Courtade}\IsCorresp}

\address{Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720 USA}

\email{courtade@berkeley.edu}

\author{\firstname{Max}  \lastname{Fathi}}

\address{Université Paris Cité and Sorbonne Université, CNRS, Laboratoire Jacques-Louis Lions and Laboratoire de Probabilit\'es, Statistique et Mod\'elisation, F-75013 Paris, France}

\address{DMA, École normale supérieure, Université PSL, CNRS, 75005 Paris, France} 

\address{Institut Universitaire de France}

\email{mfathi@lpsm.paris}

\author{\firstname{Dan}   \lastname{Mikulincer}}

\address{Department of Mathematics, MIT, Cambridge, MA 02139 USA}

\email{danmiku@mit.edu}


 \thanks{T.C. acknowledges NSF-CCF 1750430, the hospitality of the Fondation Sciences Mathématiques de Paris (FSMP) and the LPSM at Université Paris Cité. M.F. was supported by the Agence Nationale de la Recherche (ANR) Grant ANR-23-CE40-0003 (Project CONVIVIALITY). D.M. was partially supported by a Simons Investigator Award (622132, PI: Mossel)}

\CDRGrant[NSF-CCF]{1750430}
\CDRGrant[ANR]{ANR-23-CE40-0003}
\CDRGrant[Simons Investigator Award]{622132}

\subjclass{60H30, 39B62, 52A40}

\keywords{\kwd{Transport inequalities} \kwd{Gaussian inequalities} \kwd{Blaschke--Santal\'o inequality} \kwd{Martingale representations}}

\altkeywords{\kwd{Inégalités de transport} \kwd{inégalités Gaussiennes} \kwd{inégalité de Blaschke--Santaló} \kwd{représentations de martingales}}


\begin{abstract} 
A new proof of the sharp symmetrized form of Talagrand's transport-entropy inequality is given.  Compared to stochastic proofs of other Gaussian functional inequalities, the new idea here is a certain  coupling induced by time-reversed martingale representations. 
\end{abstract}

\begin{altabstract} 
Nous donnons une nouvelle preuve de la version symétrisée de l'inégalité de transport-entropie de Talagrand avec constante optimale. En comparaison avec d'autres preuves stochastiques d'inégalités fonctionnelles gaussiennes,  l'élément nouveau ici est l'utilisation d'un couplage induit par un retournement du temps sur des représentations de martingales. 
\end{altabstract}


\begin{document}

\maketitle

\section{Introduction}

We aim to  give a  short  stochastic proof of the following sharp symmetrized Talagrand inequality:
\begin{theorem}[\hspace{-.1mm}{\cite[Theorem~1.1]{MF}}]\label{thm:symmetrizedTalagrand}
For  Borel probability measures $\mu,\nu$ on $\mathbb{R}^n$ with finite second moments and $\mu$ centered,  
\begin{align} 
W_2(\mu,\nu)^2 \leq 2 D(\mu\|\gamma)  + 2 D(\nu\|\gamma), \label{eq:symmTal}
\end{align}
where $W_2$ is   2-Wasserstein distance, $D$ is  relative entropy, and $\gamma$ is the standard Gaussian measure on $\mathbb{R}^n$. 
\end{theorem}
By duality, \eqref{eq:symmTal} is  formally equivalent to the functional Blaschke--Santal\'o inequality \cite[Theorem~1.2]{LBS}, which states that if Borel functions $f,g : \mathbb{R}^n \to \mathbb{R}$ satisfy 
\begin{align}
\int_{\mathbb{R}^n} x \expe^{-f(x)}\, \dd x = 0 ~~~\mbox{and}~~~f(x) + g(y) \geq \langle x, y\rangle, ~~\forall x,y\in \mathbb{R}^n,  \notag
\end{align}
then 
\begin{align} 
\left(\int_{\mathbb{R}^n} {\expe^{-f(x)}}\, \dd x\right)\left(\int_{\mathbb{R}^n} {\expe^{-g(x)}\, \dd x}\right) \leq (2\pi)^n. \label{functional_santalo}
\end{align}
Equality holds for quadratic $f$, and $g = f^*$, its convex conjugate.   Despite the  equivalence, \eqref{eq:symmTal} may be regarded as a formal strengthening of \eqref{functional_santalo} in the sense that \eqref{functional_santalo} is recovered from Theorem \ref{thm:symmetrizedTalagrand} by weak duality: briefly,  for $f,g$ satisfying the hypotheses demanded by \eqref{functional_santalo},   take $\dd \mu(x) \propto \expe^{-f(x)} \,\dd x$ and  $\dd \nu(x) \propto \expe^{-g(x)} \,\dd x$  in \eqref{eq:symmTal} and simplify to obtain \eqref{functional_santalo}.  The reverse implication corresponds to strong duality, and is more difficult. See \cite[Theorem~11]{G07} and \cite{MF} for details.

Inequality  \eqref{functional_santalo} is a functional generalization of the earlier  Blaschke--Santal\'o inequality for the volume product of convex sets,  earlier proofs of which were accomplished by calculus of variations \cite{Santalo} and symmetrization arguments \cite{SR,MP}.  The functional form was proved in \cite{AKM} (and earlier in K. Ball's Ph.D.~thesis \cite{Ball} in a restricted setting of even functions). The original proof relied on the usual Blaschke--Santal\'o inequality applied to level sets. Lehec later gave two alternative proofs; one using induction on the dimension \cite{LBS}, and the other \cite{Leh} using the Prek\'opa--Leindler inequality and the Yao--Yao partition theorem. This last proof actually yields a more general statement, originally due to \cite{FM}, but the present work shall be restricted to the classical setting. More recently, a new semigroup proof of the inequality for even functions was established in \cite{NT} using improved hypercontractive estimates for the heat flow, and then simplified in \cite{CGNT}. Let us also mention a recent generalization to several functions under a symmetry assumption, due to Kolesnikov and Werner \cite{KW}.

Equivalence between integral inequalities of the form \eqref{functional_santalo} and transport inequalities of the form \eqref{eq:symmTal} via duality goes back to \cite{BG}, where they studied Talagrand quadratic transport-entropy inequality \cite{Tal} (which is \eqref{eq:symmTal} in the particular case $\mu = \gamma$). Duality for transport inequalities involving three measures, such as \eqref{eq:symmTal}, has been  considered in \cite{G07} and \cite[Proposition~8.2]{GL}.

 Stochastic proofs of functional inequalities, in particular using Brownian motion and Girsanov's theorem, go back to Borell's stochastic proof of the Prek\'opa--Leindler inequality \cite{Bor}. Our present work is motivated by Lehec's short stochastic proofs of various functional inequalities \cite{L}, including in particular Talagrand's transport-entropy inequality.



\section{A Stochastic Proof}

We'll work on   the  Wiener space $(\Omega, \mathcal{B}, \mathbb{P})$, where $\Omega$ is the set of continuous paths $\omega : [0,1]\to \mathbb{R}^n$ starting at 0, $\mathcal{B}$ is the usual Borel $\sigma$-algebra, and $\mathbb{P}$ is the Wiener measure.  Let $B_t(\omega) := \omega(t)$ be the coordinate process, so that $B = (B_t)_{0\leq t \leq 1}$ is a standard Brownian motion, and so is the time-reversed process $\hat{B}_t := B_1 - B_{1-t}$.    Let $\mathcal{F} = (\mathcal{F}_t)_{0 \leq t \leq 1}$ and $\mathcal{F}^+ = (\mathcal{F}^+_{t})_{0 \leq t \leq 1}$ denote the filtrations generated by $B$ and $\hat{B}$, respectively.  For each $t\in [0,1]$,  $\mathcal{F}_t$ and $\mathcal{F}^+_{1-t}$ are complementary, in the sense that they are independent and $\mathcal{B} =\sigma( \mathcal{F}_t \cup \mathcal{F}_{1-t}^+)$.  Henceforth, $\|\cdot\|$ denotes the $\ell^2$ norm, and $\id$ denotes the identity matrix.

Apart from standard facts in stochastic calculus, we'll need two lemmas.  The first is a  variational representation of entropy, obtained as a consequence of Girsanov's theorem; it has   been  applied to study rigidity and stability of various functional inequalities (see, e.g.,  \cite{ACZ, EM, M}).
\begin{lemma}[\cite{EM}]\label{lem:varRep}
For  a centered probability measure $\mu$  on $\mathbb{R}^n$ with finite second moments, we have 
\begin{align}
D(\mu \| \gamma) = \inf_{F} \frac{1}{2}\int_{0}^1 \frac{\EE[\|F_t - \id\|^2 ]}{1-t}\,\dd t, \label{repEntropy} 
\end{align}
where the infimum is over    $\mathcal{F}$-adapted $\mathbb{R}^{n\times n}$-valued processes $F = (F_t)_{0 \leq t \leq 1}$ such that $\int_0^1 F_t\, \dd B_t \sim \mu$.  
\end{lemma}
\begin{remark}
The same representation holds if we consider $\mathcal{F}^+$-adapted     $F$ with $\int_0^1 F_t\, \dd \hat{B}_t \sim \mu$.
\end{remark}
Stochastic proofs of several other  functional inequalities use representation formulas for the entropy and linear couplings of Brownian motions (cf.~\cite{EM,L}). Our proof will similarly rely on the  representation formula \eqref{repEntropy} for the entropy\footnote{The representation \eqref{repEntropy}    is not the same as that used in \cite{L}. However, it is derived from \cite[Theorem~4]{L} by combination with the martingale representation theorem.}, but makes use of a  new coupling induced by time-reversed martingale representations.    The next lemma is the crucial new ingredient; it relates   martingale representations in terms of $(B_t)_{0\leq t \leq 1}$ and its time-reversal $(\hat{B}_t)_{0\leq t \leq 1}$. 
\begin{lemma}\label{lem:integralInequality}
If $X \in L^2(\Omega, \mathcal{B}, \mathbb{P})$ is a    $\mathbb{R}^n$-valued random vector with    martingale representations
\[
X = \int_0^1 F_t\, \dd B_t = \int_{0}^1 G_t \,\dd \hat{B}_t,
\]
then 
\begin{align}
   \int_{t}^1 \EE[\|F_s-\id\|^2]\, \dd s \geq  \int_0^{1-t} \EE[\|G_{s}-\id \|^2]\, \dd s , ~~~\forall 0 \leq t \leq 1.  \label{eq:tailBounds}
\end{align}  
\end{lemma}
\begin{proof}
By the Pythagorean theorem, convexity, and  independence of $\mathcal{F}_t$ and $\mathcal{F}^+_{1-t}$, we have 
\begin{align*}
\EE[ \|X\|^2] - \EE[\|\EE[X|\mathcal{F}_t]\|^2] = \EE[\|X - \EE[X|\mathcal{F}_{t}]\|^2]   \geq    \EE[\|\EE[X- \EE[X|\mathcal{F}_{t}]|\mathcal{F}^+_{1-t}]\|^2]= \EE[\|\EE[X|\mathcal{F}^+_{1-t}]\|^2]. % \label{eq:Pythagorean}
\end{align*}
Since $\EE[X|\mathcal{F}_t] = \int_0^t F_s \, \dd B_s$ and $\EE[X|\mathcal{F}^+_{1-t}] = \int_0^{1-t} G_s \, \dd \hat{B}_s$, three applications of It\^o's isometry give 
\begin{align}
     \int_{t}^1 \EE[\|F_s\|^2]\, \dd s = \EE[ \|X\|^2] - \EE[\|\EE[X|\mathcal{F}_t]\|^2] \geq  \EE[\|\EE[X|\mathcal{F}^+_{1-t}]\|^2] = \int_0^{1-t} \EE[\|G_s\|^2]\, \dd s  , ~~~\forall 0 \leq t \leq 1.   \notag % \label{eq:Pythagorean}
\end{align}
Next,  by applying It\^o's isometry to each martingale representation of $X$, we find
\[
\int_t^1 \EE[\tr(F_s)] \, \dd s = \EE[\langle X, B_1- B_t\rangle ] = \EE[\langle X ,\hat{B}_{1-t}  \rangle] = \int_0^{1-t} \EE[\tr(G_s)] \, \dd s,   ~~~\forall 0 \leq t \leq 1.
\]
Combining  the previous   two observations   gives \eqref{eq:tailBounds}.   
\end{proof}
 
\begin{proof}[Proof of Theorem \ref{thm:symmetrizedTalagrand}] The inequality is invariant with respect to translations of $\nu$, so we may also assume $\nu$ is  centered.  
 Let $F = (F_t)_{0 \leq t \leq 1}$ be any $\mathcal{F}$-adapted process  such that $\int_0^1 F_t \, \dd B_t \sim \mu$, and let $H = (H_t)_{0 \leq t \leq 1}$ be any $\mathcal{F}^+$-adapted process such that $\int_0^1 H_t \, \dd \hat{B}_t \sim \nu$.   Let $G = (G_t)_{0 \leq t \leq 1}$ be the martingale representation of  $\int_0^1 F_t \, \dd B_t$ in terms of the time-reversed Brownian motion $\hat{B}$; i.e., $G$ is $\mathcal{F}^+$-adapted, satisfying $\int_0^1 G_t \, \dd \hat{B}_t =\int_0^1 F_t \, \dd B_t \sim \mu$.   By the Tonelli theorem and Lemma \ref{lem:integralInequality}, we have  
      \begin{align*}
      \int_0^1 \frac{\EE[\|G_{s}-\id\|^2]}{s} \, \dd s&= \int_0^1  \EE[\|G_{s}-\id \|^2] \, \dd s +  \int_0^1 \frac{1}{(1-t)^2} \left( \int_0^{1-t} \EE[\|G_{s}-\id \|^2] \, \dd s \right) \, \dd t\\
      &\leq \int_0^1  \EE[\|F_s-\id \|^2] \, \dd s +  \int_0^1 \frac{1}{(1-t)^2} \left( \int_t^1 \EE[\|F_s-\id \|^2] \, \dd s \right) \, \dd t = \int_0^1 \frac{\EE[\|F_s-\id \|^2]}{1-s}\, \dd s.
   \end{align*}
By definition of $W_2$, It\^o's isometry,    convexity of $\|\cdot\|^2$, and the previous estimate, 
\begin{align*}
W_2(\mu,\nu)^2 \leq \EE \left\| \int_{0}^1 (G_t -H_t)\, \dd \hat{B}_t  \right\|^2 
=  \int_{0}^1 \EE [\|G_t -H_{t}\|^2] \, \dd t
 &\leq \int_{0}^1 \frac{\EE [\|G_t - \id\|^2]}{t} \, \dd t + \int_{0}^1 \frac{\EE [\|H_{t} - \id\|^2]}{1-t}\, \dd t
 \\ &\leq   \int_0^1 \frac{\EE[\|F_t-\id \|^2]}{1-t}\, \dd t+  \int_{0}^1 \frac{\EE [\|H_t - \id\|^2]}{1-t} \, \dd t.
\end{align*}
With the help of  Lemma  \ref{lem:varRep}, optimizing over $F$ and $H$  completes the proof . \end{proof}



\section{Remarks on the Approach}
\subsection{Equality cases} The  equality cases for \eqref{eq:symmTal} are also evident from the given proof.  Indeed, if $D(\mu\|\gamma) <\infty$, then the infimum in \eqref{repEntropy} is a.s.-uniquely achieved by an $\mathcal{F}$-adapted process $F = (F_t)_{0 \leq t \leq 1}$.  Defining $X := \int_0^1 F_t \, \dd B_t\sim \mu$ for this  particular $F$, equality in \eqref{eq:symmTal} implies equality in \eqref{eq:tailBounds} for a.e.~$t\in[0,1]$, which requires that $X - \EE[X|\mathcal{F}_t]$ is $\mathcal{F}^+_{1-t}$-measurable for a.e.~$t\in [0,1]$.  By Proposition \ref{prop:Gaussian} below,  this ensures $X\sim \mu$ is Gaussian.  By symmetry, any extremal $\nu$ is also Gaussian, and  explicit computation shows that  $\mu,\nu$ are extremizers in \eqref{eq:symmTal} if and only if   $\mu = N(0,C)$ and $\nu = N(\theta,C^{-1})$ for  some $\theta\in \mathbb{R}^n$ and positive  definite $C\in \mathbb{R}^{n\times n}$.  
\begin{proposition}\label{prop:Gaussian}
Let $X\in L^2(\Omega, \mathcal{B},\mathbb{P})$ admit martingale representation $X = \int_0^1 F_t \, \dd B_t$.  If $X - \EE[X|\mathcal{F}_t]$ is $\mathcal{F}^+_{1-t}$-measurable for a.e.~$t\in [0,1]$, then $X$ is Gaussian. 
\end{proposition}
\begin{proof}
Define $M_t := \int_{0}^t F_s \, \dd B_s$.  The hypothesis is equivalent to requiring that $(M_1 - M_t)$ is $\mathcal{F}^+_{1-t}$-measurable for each $t\in \mathcal{D}$, where $\mathcal{D}$ is dense in $[0,1]$.  Fix  any $s,t\in \mathcal{D}$, with $s\leq t$.  Since  $(M_1 - M_t)$ is $\mathcal{F}^+_{1-t}$-measurable by hypothesis, and $(M_t - M_s)$ is $\mathcal{F}_{t}$-measurable by definition,  complementarity ensures $(M_1 - M_t)$ and $(M_t - M_s)$ are independent.  Iterating this procedure on the $(M_1 - M_t)$ term allows us to conclude that $(M_t)_{0 \leq t \leq 1}$ has independent increments, provided the endpoints of the increments are in $\mathcal{D}$.  Since $X\in L^2(\Omega, \mathcal{B},\mathbb{P})$, a version of $(M_t)_{0 \leq t \leq 1}$ admits continuous  sample paths, and we conclude by density of $\mathcal{D}$ that $(M_t)_{0 \leq t \leq 1}$ has (square-integrable) independent increments generally, and is thus a Gaussian process.
\end{proof}




\subsection{Importance of the coupling induced by time-reversal} With the proof of Theorem \ref{thm:symmetrizedTalagrand} in hand and the equality cases characterized, we  highlight the importance of the coupling based on time-reversal. Following previous stochastic proofs of functional inequalities, one could  appeal to  martingale representations   $\int_0^1 F_t \, \dd B^1_t \sim \mu$ and $\int_0^1 G_t \, \dd B^2_t \sim \nu$ with linearly coupled Brownian motions $B^1$ and $B^2$ (equivalently, Brownian motions $B^1$ and $B^2$ adapted to a common filtration) to couple $\mu$ and $\nu$.   This approach cannot work, as we now explain. 		 
	
	Working in dimension $n=1$ for simplicity, recall that when $\mu = N(0,\alpha)$ with $\alpha>0$, the minimizer $F$ in \eqref{repEntropy} has an  explicit expression  (e.g., \cite[Sec.~2]{EM}).  In particular,  
				\begin{align*}
	\int F_t \, \dd B^1_t \sim \mu \mbox{~and~} D(\mu \| \gamma) =   \frac{1}{2}\int_{0}^1 \frac{\EE[\|F_t - \id\|^2 ]}{1-t}\, \dd t ~~\to~~  	F_t = \frac{\alpha}{1-t + \alpha t}.
	\end{align*}
Likewise, for $\nu = N(0,\alpha^{-1})$,  the ``optimal''   representation of $\nu$ with respect to $B^2$ satisfies   
	\[
 \int_0^1 G_t \, \dd B^2_t \sim \nu \mbox{~and~} D(\nu \| \gamma) =   \frac{1}{2}\int_{0}^1 \frac{\EE[\|G_t - \id\|^2 ]}{1-t}\, \dd t ~~\to~~ G_t = \frac{1}{ \alpha(1-t) +  t}.
\] 
Since $B^1$ and $B^2$ are linearly coupled standard Brownian motions, we can write 
\begin{align}
\begin{bmatrix}
B^1_t \\ B^2_t
\end{bmatrix} = \begin{bmatrix}
1 & \sigma \\ \sigma & 1
\end{bmatrix}^{1/2} B_t, ~~~0 \leq t \leq 1,\label{linCoupling}
\end{align}
for some $|\sigma| \leq 1$, where $(B_t)_{0 \leq t \leq 1}$ is a 2-dimensional standard Brownian motion.  This construction induces a  coupling $\pi_{\sigma}$  of $X := \int_0^1 F_t \, \dd B^1_t\sim \mu$ and $Y := \int_0^1 G_t \, \dd B^2_t\sim \nu$ satisfying
\begin{align*}
\EE_{\pi_{\sigma}} \|X-Y\|^2 %= \EE \left\| \int_0^1 F_t \, \dd B^1_t- \int_0^1 G_t \, \dd B^2_t  \right\|^2 &
= \int_0^1 \left(  F_t^2 + G_t^2 - 2 \sigma F_t G_t \right) \, \dd t = \begin{cases}
 \left(\alpha + \frac{1}{\alpha}\right)  -   \frac{4 \sigma \log \alpha}{ \left(\alpha - \frac{1}{\alpha}\right) } & \mbox{if~}\alpha \neq 1\\
 2(1-\sigma) & \mbox{if~}\alpha = 1,
 \end{cases}
\end{align*}
where we made use of It\^o's isometry and \eqref{linCoupling}.  A simple calculation  reveals that 
\[
\min_{\sigma : |\sigma|\leq 1} \EE_{\pi_{\sigma}} \|X-Y\|^2 \geq \alpha + \frac{1}{\alpha}  - 2 = W_2(\mu,\nu)^2 = 2 D(\mu\|\gamma) + 2 D(\nu\|\gamma), 
\]
with equality if and only if $\alpha = 1$.  So, with the exception of  the trivial case $\mu = \nu = \gamma$, the established stochastic approach to proving functional inequalities using linearly coupled Brownian motions fails to produce the requisite optimal coupling between $\mu$ and $\nu$ in all extremal cases (at least, in this implementation).  This suggests that coupling through time-reversal lends a useful new degree of freedom to the stochastic program for proving functional inequalities.  

\section*{Acknowledgments} 
The authors thank Joseph Lehec and an anonymous referee for their careful reading of the paper and suggestions for improvement.

\section*{Declaration of interests}
The authors do not work for, advise, own shares in, or receive funds from
any organization that could benefit from this article, and have declared no
affiliations other than their research organizations.


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