Log-Hessian formula and the Talagrand conjecture

N Gozlan, Xue-Mei Li, M Madiman, C. Roberto, P. -M Samson

Published 2019-07-25Version 1

In 1989, Talagrand proposed a conjecture regarding the regularization effect on integrable functions of a natural Markov semigroup on the Boolean hypercube. While this conjecture remains unresolved, the analogous conjecture for the Ornstein-Uhlenbeck semigroup was recently resolved by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of this semigroup with a new deviation inequality for log-semiconvex functions under Gaussian measure. Our first goal is to explore the validity of both these ingredients for some diffusion semigroups in R n as well as for the M/M/$\infty$ queue on the non-negative integers. Our second goal is to prove an analogue of Talagrand's conjecture for these settings, even in those cases where these ingredients are not valid.