Poincaré inequality and the Lp convergence of semi-groups

Abstract

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