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

Patrick Cattiaux (Institut de Mathématiques de Toulouse)
Arnaud Guillin (Université Blaise Pascal)
Cyril Roberto (Universités de Paris Est Marne la Vallée et de Paris 12-Val-de-Marne)

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.

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Pages: 270-280

Publication Date: June 9, 2010

DOI: 10.1214/ECP.v15-1559

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. MR2381160
  • Cattiaux, Patrick. A pathwise approach of some classical inequalities. Potential Anal. 20 (2004), no. 4, 361--394. MR2032116
  • Cattiaux, Patrick; Guillin, Arnaud. Trends to equilibrium in total variation distance. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), no. 1, 117--145. MR2500231
  • Roberto, C.; Zegarliński, B. Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J. Funct. Anal. 243 (2007), no. 1, 28--66. MR2289793
  • Villani, Cédric. Hypocoercivity. Mem. Amer. Math. Soc. 202 (2009), no. 950, iv+141 pp. ISBN: 978-0-8218-4498-4 MR2562709
  • Wang, Feng-Yu. Probability distance inequalities on Riemannian manifolds and path spaces. J. Funct. Anal. 206 (2004), no. 1, 167--190. MR2024350
  • Wu, Liming. Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition. Ann. Probab. 34 (2006), no. 5, 1960--1989. MR2271488
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