Electronic Communications in Probability

Poincaré inequality and the $L^p$ convergence of semi-groups

Patrick Cattiaux, Arnaud Guillin, and Cyril Roberto

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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

Permanent link to this document
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

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60G10: Stationary processes 60J60: Diffusion processes [See also 58J65]

Keywords
Poincaré inequality rate of convergence

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

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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