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

Dominique Bakry (LSP, Univ. Toulouse 3)
Franck Barthe (LSP, Univ. Toulouse 3)
Patrick Cattiaux (LSP, Univ. Toulouse 3)
Arnaud Guillin (LATP, Univ. Aix-Marseille 1)

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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 60-66

Publication Date: February 4, 2008

DOI: 10.1214/ECP.v13-1352

References

  • C. Ané, S. Blachère, D. Chafa"i, 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. MR2381160
  • Bobkov, S. G. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27 (1999), no. 4, 1903--1921. MR1742893
  • Cattiaux, Patrick. Hypercontractivity for perturbed diffusion semigroups. Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 609--628. MR2188585
  • 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. MR2126968
  • 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. MR2195409
  • Wu, Liming. Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172 (2000), no. 2, 301--376. MR1753178


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.