Functional inequalities for some generalised Mehler semigroups

TL;DR

This paper establishes functional inequalities like logarithmic Sobolev and Poincaré inequalities for generalized Mehler semigroups assuming an invariant measure, leading to new integrability results.

Contribution

It introduces new functional inequalities for generalized Mehler semigroups under invariant measure assumptions, expanding understanding of their analytical properties.

Findings

Proves logarithmic Sobolev inequalities for generalized Mehler semigroups.

Establishes Poincaré inequalities under invariant measure assumptions.

Derives integrability properties of exponential functions with respect to the invariant measure.

Abstract

We consider generalised Mehler semigroups and, assuming the existence of an associated invariant measure $\sigma$, we prove functional integral inequalities with respect to $\sigma$, such as logarithmic Sobolev and Poincar\'{e} type. Consequently, some integrability properties of exponential functions with respect to $\sigma$ are deduced.