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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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

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

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

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Zentralblatt MATH identifier

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]

Poincaré inequality rate of convergence

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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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  • C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer. Sur les inégalités de Sobolev logarithmiques, volume 10 of Panoramas et Synthèses. Société Mathématique de France, Paris, 2000.
  • Bakry, Dominique; Cattiaux, Patrick; Guillin, Arnaud. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (2008), no. 3, 727–759.
  • Cattiaux, Patrick. A pathwise approach of some classical inequalities. Potential Anal. 20 (2004), no. 4, 361–394.
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  • Roberto, C.; ZegarliÅ„ski, B. Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J. Funct. Anal. 243 (2007), no. 1, 28–66.
  • Villani, Cédric. Hypocoercivity. Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141 pp. ISBN: 978-0-8218-4498-4
  • Wang, Feng-Yu. Probability distance inequalities on Riemannian manifolds and path spaces. J. Funct. Anal. 206 (2004), no. 1, 167–190.
  • Wu, Liming. Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition. Ann. Probab. 34 (2006), no. 5, 1960–1989.