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

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