Electronic Communications in Probability

A simple proof of the Poincaré inequality for a large class of probability measures

Dominique Bakry, Franck Barthe, Patrick Cattiaux, and Arnaud Guillin

Full-text: Open access

Abstract

Abstract. We give a simple and direct proof of the existence of a spectral gap under some Lyapunov type condition which is satisfied in particular by log-concave probability measures on $\mathbb{R}^n$. The proof is based on arguments introduced in Bakry and al, but for the sake of completeness, all details are provided.

Article information

Source
Electron. Commun. Probab., Volume 13 (2008), paper no. 7, 60-66.

Dates
Accepted: 4 February 2008
First available in Project Euclid: 6 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1465233434

Digital Object Identifier
doi:10.1214/ECP.v13-1352

Mathematical Reviews number (MathSciNet)
MR2386063

Zentralblatt MATH identifier
1186.26011

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60G10: Stationary processes 60J60: Diffusion processes [See also 58J65]

Keywords
Lyapunov functions Poincaré inequality log-concave measure

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

Citation

Bakry, Dominique; Barthe, Franck; Cattiaux, Patrick; Guillin, Arnaud. A simple proof of the Poincaré inequality for a large class of probability measures. Electron. Commun. Probab. 13 (2008), paper no. 7, 60--66. doi:10.1214/ECP.v13-1352. https://projecteuclid.org/euclid.ecp/1465233434


Export citation

References

  • 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.
  • Bobkov, S. G. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999), no. 4, 1903–1921.
  • Cattiaux, Patrick. Hypercontractivity for perturbed diffusion semigroups. Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 609–628.
  • P. Cattiaux, A. Guillin, F. Y. Wang, and L. Wu. Lyapunov conditions for logarithmic Sobolev and super Poincaré inequality. Available on Math. ArXiv 0712.0235., 2007.
  • Fougères, Pierre. Spectral gap for log-concave probability measures on the real line. Séminaire de Probabilités XXXVIII, 95–123, Lecture Notes in Math., 1857, Springer, Berlin, 2005.
  • Ledoux, Michel. Spectral gap, logarithmic Sobolev constant, and geometric bounds. Surveys in differential geometry. Vol. IX, 219–240, Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004.
  • Wu, Liming. Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172 (2000), no. 2, 301–376.