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.
Citation
Patrick Cattiaux. Arnaud Guillin. Cyril Roberto. "Poincaré inequality and the $L^p$ convergence of semi-groups." Electron. Commun. Probab. 15 270 - 280, 2010. https://doi.org/10.1214/ECP.v15-1559
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