Dimension dependent hypercontractivity for Gaussian kernels

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein–Uhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton–Jacobi equation. Hypercontractive bounds on the Ornstein–Uhlenbeck semigroup driven by a non-diffusive Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.

[1]  L. Gross.  LOGARITHMIC SOBOLEV INEQUALITIES. , 1975 .

[2]  C. Villani,et al.  Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality , 2000 .

[3]  Logarithmic Sobolev inequalities: Regularizing effect of Lévy operators and asymptotic convergence in the Lévy–Fokker–Planck equation , 2008, 0809.2654.

[4]  Djalil CHAFAÏ.  Entropies, convexity, and functional inequalities : On Phi-entropies and Phi-Sobolev inequalities , 2004 .

[5]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[6]  D. Bakry,et al.  Around Nash Inequalities , 2010 .

[7]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[8]  Subelliptic Li-Yau estimates on three dimensional model spaces , 2008, 0806.2547.

[9]  tballest.  Séminaire de probabilités , 2013 .

[10]  P. Meyer,et al.  Sur les inegalites de Sobolev logarithmiques. I , 1982 .

[11]  G. Barles.  Solutions de viscosité des équations de Hamilton-Jacobi , 1994 .

[12]  On logarithmic Sobolev inequalities for continuous time random walks on graphs , 2000 .

[13]  Michel Ledoux,et al.  A logarithmic Sobolev form of the Li-Yau parabolic inequality , 2006 .

[14]  D. Bakry,et al.  Weighted Nash Inequalities , 2010, 1004.3456.

[15]  I. Gentil.  The general optimal Lp-Euclidean logarithmic Sobolev inequality by Hamilton–Jacobi equations , 2003 .

[16]  Christian L'eonard,et al.  Transport Inequalities. A Survey , 2010, 1003.3852.

[17]  C. Villani.  The founding fathers of optimal transport , 2009 .

[18]  D. Applebaum.  Lévy Processes and Stochastic Calculus: Preface , 2009 .

[19]  M. Talagrand.  Transportation cost for Gaussian and other product measures , 1996 .

[20]  Dominique Bakry,et al.  Functional Inequalities for Markov semigroups , 2009 .

[21]  C. Villani.  Optimal Transport: Old and New , 2008 .

[22]  E. Lieb.  Gaussian kernels have only Gaussian maximizers , 1990 .

[23]  I. Gentil.  Ultracontractive bounds on Hamilton–Jacobi solutions , 2002 .

[24]  S. Bobkov,et al.  Exponential Integrability and Transportation Cost Related to Logarithmic Sobolev Inequalities , 1999 .

[25]  S. Bobkov,et al.  Hypercontractivity of Hamilton-Jacobi equations , 2001 .

[26]  Liming Wu,et al.  A new modified logarithmic Sobolev inequality for Poisson point processes and several applications , 2000 .