Hypercontractivity of Hamilton - Jacobi Equations

S.G. Bobkov, I. Gentil, M. Ledoux

2002, v.8, Issue 2, 233-235

ABSTRACT

Following the equivalence between logarithmic Sobolev inequalities and hypercontractivity showed by L. Gross, we prove that logarithmic Sobolev inequalities are related similarly to hypercontractivity of solutions of Hamilton - Jacobi equations. Given a bounded Lipschitz function $f$, solutions of the Hamilton - Jacobi initial value problem $$ \left \{ \begin{array}{rcll} \frac{\partial v}{\partial t} + \frac{1}{2} \, |\nabla v|^2 & = & 0& {\rm in}\,\, R^n \times(0,\infty ), v & = & f& {\rm on} \,\, R^n \times \{t=0\}, \end{array} \right. $$ are described by the Hopf - Lax representation formula as the infimum-convolution of $f$ with the quadratic cost $$ Q_t f(x) = \inf _{y \in R^n} \big [ f(y) + \frac{1}{2t} \, |x-y|^2 \big ], \quad t>0, \, x \in R^n. $$ Our main result is the following theorem. Let $\mu $ be a probability measure on the Borel sets of $R^n$ absolutely continuous with respect to Lebesgue measure such that for some $\rho >0$ and all smooth enough functions $\varphi $ on $R^n$ with $\int \varphi ^2 d\mu =1$, \begin{equation} \label{(1)} \rho \int \varphi ^2 \log \varphi ^2 d\mu \leq 2 \int |\nabla \varphi |^2 d\mu . \end{equation} Then for every bounded measurable function $f$ on $R^n$, every $t > 0$ and every $a \in R$, \begin{equation} \label{(2)} {\big\| \,{\rm e}^{Q_t f} \big\|}_{a + \rho t} \leq {\big\|\, {\rm e}^f\big\|}_a \end{equation} (where the norms are understood with respect to $\mu $). Conversely, if (\ref{(2)}) holds for all $t> 0$ and some $a \not= 0$, then the logarithmic Sobolev inequality (\ref{(1)}) holds. When $a=0$, (\ref{(2)}) actually amounts to the infimum-convolution inequality $$ \int {\rm e}^{\,\rho \,Q_1 f} d\mu \leq {\rm e}^{\,\rho \int f d\mu }$$ holding for every bounded (or integrable) function $f$. This inequality is known to be the Monge - Kantorovitch - Rubinstein dual version of the transportation cost inequality $$ \rho \, W_2(\mu ,\nu)^2 \leq H(\nu \, | \, \mu ) $$ holding for all probability measures $\nu$ absolutely continuous with respect to $\mu $ with Radon - Nikodym derivative $d\nu / d\mu $. Here $W_2$ is the Wasserstein distance with quadratic cost $$ W_2 (\mu , \nu)^2 = \inf \int \int {\frac{1}{2}} \, |x-y|^2 d\pi (x,y) $$ where the infimum is running over all probability measures $\pi $ on $R^n \times R^n$ with respective marginals $\mu $ and $\nu$ and $H(\nu \, | \, \mu )$ is the relative entropy, or informational divergence, of $\nu$ with respect to $\mu $. This approach thus provides a clear view of the connection between logarithmic Sobolev inequalities and transportation cost inequalities investigated recently by F. Otto and C. Villani. Extensions to the Riemannian setting and applications to transportation cost and concentration inequalities, HWI inequalities and isoperimetry complete this work.

Keywords: hypercontractivity,logarithmic Sobolev inequality,transportation costinequality,Hamilton - Jacobi equations

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