LOG-HESSIAN FORMULA AND THE TALAGRAND CONJECTURE

Abstract : 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/∞ 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.
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Submitted on : Wednesday, July 24, 2019 - 9:30:44 AM
Last modification on : Saturday, July 27, 2019 - 1:21:10 AM

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  • HAL Id : hal-02192650, version 1
  • ARXIV : 1907.10896

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N Gozlan, Xue-Mei Li, M Madiman, C. Roberto, P.-M Samson. LOG-HESSIAN FORMULA AND THE TALAGRAND CONJECTURE. 2019. ⟨hal-02192650⟩

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