Poincar\'e inequality and the Lp convergence of semi-groups

TL;DR

This paper establishes that for certain symmetric diffusion processes, the Poincaré inequality is equivalent to exponential convergence of their semi-groups in various L^p spaces, with some extensions to non-symmetric cases.

Contribution

It proves the equivalence between Poincaré inequality and exponential semi-group convergence in L^p spaces for symmetric Markov processes, extending to non-symmetric stationary cases.

Findings

Poincaré inequality implies exponential convergence in L^p spaces.

Equivalence between Poincaré inequality and semi-group convergence.

Extension to non-symmetric stationary processes.

Abstract

We prove that for symmetric Markov processes of diffusion type admitting a "carr\'e du champ", the Poincar\'e inequality is equivalent to the exponential convergence of the associated semi-group in one (resp. all) $\L^p(\mu)$ spaces for $p\in (1,+\infty)$. Part of this result extends to the stationary non necessarily symmetric situation.