Poincaré inequality and the Lp convergence of semi-groups

International audienceWe 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 $p\in (1,+\infty)$. Part of this result extends to the stationary non necessarily symmetric situation