## Electronic Communications in Probability

### 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.

#### Article information

Source
Electron. Commun. Probab., Volume 15 (2010), paper no. 25, 270-280.

Dates
Accepted: 9 June 2010
First available in Project Euclid: 6 June 2016

https://projecteuclid.org/euclid.ecp/1465243968

Digital Object Identifier
doi:10.1214/ECP.v15-1559

Mathematical Reviews number (MathSciNet)
MR2661206

Zentralblatt MATH identifier
1223.26037

Rights

#### Citation

Cattiaux, Patrick; Guillin, Arnaud; Roberto, Cyril. Poincaré inequality and the $L^p$ convergence of semi-groups. Electron. Commun. Probab. 15 (2010), paper no. 25, 270--280. doi:10.1214/ECP.v15-1559. https://projecteuclid.org/euclid.ecp/1465243968

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