Poincaré inequality and the $L^p$ 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) $L^p(\mu)$ spaces for $1 < p < \infty$. We also give the optimal rate of convergence. Part of these results extends to the stationary, not necessarily symmetric situation.