# Log-Hessian and Deviation Bounds for Markov Semi-Groups, and Regularization Effect in $L^1$

## TL;DR

This paper investigates the regularization effects of Markov semi-groups, extending known inequalities and deviation bounds from Gaussian settings to more general diffusions, queues, and Laguerre semi-groups, revealing new regularization phenomena.

## Contribution

It generalizes log-Hessian and deviation bounds to various semi-groups beyond Gaussian, and establishes a one-dimensional regularization effect in broader contexts.

## Key findings

- Log-Hessian bounds hold for certain diffusion semi-groups.
- Deviation inequalities for log-semi-convex functions are established.
- A one-dimensional regularization effect is proven in new settings.

## Abstract

It is well known that some important Markov semi-groups have a "regularization effect" -- as for example the hypercontractivity property of the noise operator on the Boolean hypercube or the Ornstein-Uhlenbeck semi-group on the real line, which applies to functions in $L^p$ for $p>1$. Talagrand had conjectured in 1989 that the noise operator on the Boolean hypercube has a further subtle regularization property for functions that are just integrable, but this conjecture remains open. Nonetheless, the Gaussian analogue of this conjecture was proven in recent years by Eldan-Lee and Lehec, by combining an inequality for the log-Hessian of the Ornstein-Uhlenbeck semi-group with a new deviation inequality for log-semi-convex functions under Gaussian measure. In this work, we explore the question of how much more general this phenomenon is. Specifically, our first goal is to explore the validity of both these ingredients for some diffusion semi-groups in $\mathbb{R}^n$, as well as for the $M/M/\infty$ queue on the non-negative integers and the Laguerre semi-groups on the positive real line. Our second goal is to prove a one-dimensional regularization effect for these settings, even in those cases where these ingredients are not valid.