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  • Print publication year: 2014
  • Online publication date: August 2014

11 - Functional inequalities via Lyapunov conditions




We review here some recent results by the authors, and various coauthors, on (weak, super) Poincaré inequalities, transportation-information inequalities or logarithmic Sobolev inequality via a quite simple and efficient technique: Lyapunov conditions.

Introduction and main concepts

Lyapunov conditions appeared a long time ago. They were particularly well fitted to deal with the problem of convergence to equilibrium for Markov processes; see [23, 38–40] and references therein. They also appeared earlier in the study of large and moderate deviations for empirical functionals of Markov processes (for examples, see Donsker and Varadhan [21, 22], Kontoyaniis and Meyn [33, 34], Wu [47, 48], Guillin [28, 29]), for solving the Poisson equation [24].

Their use to obtain functional inequalities is however quite recent, even if one may afterwards find hint of such an approach in Deuschel and Stroock [19] or Kusuocka and Stroock [35]. The present authors and coauthors have developed a methodology that has been successful for various inequalities: Lyapunov–Poincaré inequalities [4], Poincaré inequalities [3], transportation inequalities for Kullback information [17] or Fisher information [32], super Poincaré inequalities [16], weighted and weak Poincaré inequalities [13], or [18] for super weighted Poincaré inequalities. We finally refer to the forthcoming book [15] for a complete review. For more references on the various inequalities introduced here we refer to [1, 2, 36, 46]. The goal of this short review is to explain the methodology used in these papers and to present various general sets of conditions for this panel of functional inequalities. The proofs will of course be only schemed and we will refer to the original papers for complete statements.

[1] 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 Syntheses. Societe Mathematique de France, Paris, 2000.
[2] D., Bakry. L'hypercontractivite et son utilisation en theorie des semigroupes. In Lectures on Probability theory. École d'ete de Probabilites de St-Flour 1992, volume 1581 of Lecture Notes in Mathematics, pages 1-114. Springer, Berlin, 1994.
[3] D., Bakry, F., Barthe, P., Cattiaux, and A., Guillin. A simple proof of the Poincaré inequality for a large class of probability measures. Électron. Commun. in Prob., 13:60-66, 2008.
[4] D., Bakry, P., Cattiaux, and A., Guillin. Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Func. Anal., 254:727-759, 2008.
[5] D., Bakry and M., Emery. Diffusions hypercontractives. In Seminaire de probabilites, XIX, 1983/84, volume 1123 of Lecture Notes in Mathematics, pages 177-206. Springer, Berlin, 1985.
[6] F., Barthe, P., Cattiaux, and C., Roberto. Concentration for independent random variables with heavy tails. AMRX, 2005(2):39–60, 2005.
[7] F., Barthe, P., Cattiaux, and C., Roberto. Isoperimetry between exponential and Gaussian. Électron. J. Prob., 12:1212-1237, 2007.
[8] S.G., Bobkov. Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Prob., 27(4):1903–1921, 1999.
[9] S.G., Bobkov, I., Gentil, and M., Ledoux. Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pu. Appl., 80(7):669–696, 2001.
[10] S.G., Bobkov and F., Gotze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal., 163(1):1–28, 1999.
[11] S.G., Bobkov and M., Ledoux. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab., 37(2):403–427, 2009.
[12] P., Cattiaux. A pathwise approach of some classical inequalities. Potential Anal., 20:361-394, 2004.
[13] P., Cattiaux, N., Gozlan, A., Guillin, and C., Roberto. Functional inequalities for heavy tailed distributions and applications to isoperimetry. Preprint, 2008.
[14] P., Cattiaux and A., Guillin. On quadratic transportation cost inequalities. J. Math. Pures Appi., 88(4):341–361, 2006.
[15] P., Cattiaux and A., Guillin. Long time behavior of Markov processes and functional inequalities: Lyapunov conditions approach. Book in preparation, 2014.
[16] P., Cattiaux, A., Guillin, F.Y., Wang, and L. Wu. Lyapunov conditions for super Poincaré inequalities. J. Funct. Anal., 256(6):1821–1841, 2009.
[17] P., Cattiaux, A., Guillin, and L., Wu. A note on Talagrand transportation inequality and logarithmic Sobolev inequality. Probab. Theory Relat. Fields, 148:285-304, 2010.
[18] P., Cattiaux, A., Guillin, and L., Wu. Some remarks on weighted logarithmic Sobolev inequality. Indiana Univ. Math. J., 60(6):1885-1904, 2011.
[19] J.D., Deuschel and D.W., Stroock. Large Deviations, volume 137 of Pure and Applied Mathematics. Academic Press Inc., Boston, MA, 1989.
[20] H., Djellout, A., Guillin, and L., Wu. Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab., 32(3B):2702-2732, 2004.
[21] M.D., Donsker and S.R.S., Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. III. Comm. PureAppl. Math., 29(4):389–461, 1976.
[22] M.D., Donsker and S.R.S., Varadhan. Asymptotic evaluation of certain Markov process expectations for large time. IV. Comm. PureAppl. Math., 36(2):183–212, 1983.
[23] R., Douc, G., Fort, and A., Guillin. Subgeometric rates of convergence of f -ergodic strong Markov processes. Stochastic Process. Appl., 119(3):897–923, 2009.
[24] P.W., Glynn and S.P., Meyn. A Liapounov bound for solutions of the Poisson equation. Ann. Probab., 24(2):916–931, 1996.
[25] N., Gozlan. Poincaré inequalities and dimension free concentration of measure. Preprint, 2007.
[26] N., Gozlan and C., Leonard. Transportation-information inequalities. Preprint, 2009.
[27] L., Gross. Logarithmic Sobolev inequalities. Am. J. Math., 97(4):1061–1083, 1975.
[28] A., Guillin. Moderate deviations of inhomogeneous functionals of Markov processes and application to averaging. Stochastic Process. Appl., 92(2):287–313, 2001.
[29] A., Guillin. Averaging principle of SDE with small diffusion: moderate deviations. Ann. Probab., 31(1):413–443, 2003.
[30] A., Guillin, A., Joulin, C., Leonard, and L., Wu. Transportation-information inequalities for Markov processes III. Preprint, 2010.
[31] A., Guillin, C., Leonard, L., Wu, and F.Y., Wang. Transportation-information inequalities for Markov processes II. Preprint, 2009.
[32] A., Guillin, C., Leonard, L., Wu, and N., Yao. Transportation-information inequalities for Markov processes. Probab. Theory Relat. Fields, 144(3-4):669-695, 2009.
[33] I., Kontoyiannis and S.P., Meyn. Spectral theory and limit theorems for geometrically ergodic Markov processes. Ann. Appl. Probab., 13(1):304–362, 2003.
[34] I., Kontoyiannis and S. P., Meyn. Large deviations asymptotics and the spectral theory of multiplicatively regular Markov processes. Électron. J. Probab., 10(3):61–123 (electronic), 2005.
[35] S., Kusuoka and D., Stroock. Some boundedness properties of certain stationary diffusion semigroups. J. Func. Anal., 60:243-264, 1985.
[36] M., Ledoux. The Concentration of Measure Phenomenon, volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2001.
[37] K., Marton. Bounding d-distance by informational divergence: a method to prove measure concentration. Ann. Prob., 24:857-866, 1996.
[38] S.P., Meyn and R.L., Tweedie. Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag London Ltd. London, 1993.
[39] S.P., Meyn and R.L., Tweedie. Stability of Markovian processes II: continuous-time processes and sampled chains. Adv. Appl. Probab., 25:487-517, 1993.
[40] S.P., Meyn and R.L., Tweedie. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes. Adv. Appl. Probab., 25:518-548, 1993.
[41] F., Otto and C., Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal., 173(2):361–400, 2000.
[42] M., Rockner and F.Y., Wang. Weak Poincaré inequalities and L2-convergence rates of Markov semigroups. J. Funct. Anal., 185(2):564–603, 2001.
[43] M., Talagrand. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal., 6:587-600, 1996.
[44] C., Villani. Optimal Transport: Old and New, volume 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009.
[45] F.Y., Wang. Functional inequalities for empty essential spectrum. J. Funct. Anal., 170(1):219–245, 2000.
[46] F.Y., Wang. Functional Inequalities, Markov Processes and Spectral Theory. Science Press, Beijing, 2005.
[47] L., Wu. Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems. Stochastic Process. Appl., 91(2):205–238, 2001.
[48] L., Wu. Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Relat. Fields, 128(2):255–321, 2004.