Hypercontractivity for Markov Semigroups

TL;DR

This paper systematically studies hypercontractivity in Orlicz spaces for Markov semigroups related to diffusions, establishing explicit conditions and functional inequalities that characterize this property.

Contribution

It introduces a family of Orlicz functions and proves their hypercontractivity is equivalent to specific functional inequalities, advancing understanding of Markov semigroup behavior.

Findings

Explicit construction of Orlicz functions for hypercontractivity

Equivalence between hypercontractivity and functional inequalities

Application to homogeneous and non-homogeneous diffusions

Abstract

We investigate in a systematic way hypercontractivity property in Orlicz spaces for Markov semi-groups related to homogeneous and non homogeneous diffusions in $\mathbb{R}^{n}$. We provide an explicit construction of a family of Orlicz functions for which we prove that the associated hypercontractivity property is equivalent to a suitable functional inequality.