Advertisement

Semi Log-Concave Markov Diffusions

  • P. Cattiaux
  • A. Guillin
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)

Abstract

In this paper we intend to give a comprehensive approach of functional inequalities for diffusion processes under various “curvature” assumptions. One of them coincides with the usual Γ 2 curvature of Bakry and Emery in the case of a (reversible) drifted Brownian motion, but differs for more general diffusion processes. Our approach using simple coupling arguments together with classical stochastic tools, allows us to obtain new results, to recover and to extend already known results, giving in many situations explicit (though non optimal) bounds. In particular, we show new results for gradient/semigroup commutation in the log concave case. Some new convergence to equilibrium in the granular media equation is also exhibited.

Keywords

Functional inequalities Transport inequalities Diffusion processes, Coupling Convergence to equilibrium 

References

  1. 1.
    C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto, G. Scheffer, Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, vol. 10 (Société Mathématique de France, Paris, 2000)Google Scholar
  2. 2.
    D. Bakry, F. Barthe, P. Cattiaux, A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures. Electon. Commun. Probab. 13, 60–66 (2008)MathSciNetzbMATHGoogle Scholar
  3. 3.
    D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254, 727–759 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Bakry, I. Gentil, L. Ledoux, On Harnack inequalities and optimal transportation. Preprint, available on ArXiv (2012)Google Scholar
  5. 5.
    D. Bakry, I. Gentil, L. Ledoux, Analysis and Geometry of Markov diffusion operators, Springer, Grundlehren der mathematischen Wissenschaften, Vol. 348 (2014)Google Scholar
  6. 6.
    K. Ball, F. Barthe, A. Naor, Entropy jumps in the presence of a spectral gap. Duke Math. J. 119, 41–63 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    S.G. Bobkov, Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27(4), 1903–1921 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    S.G. Bobkov, Spectral gap and concentration for some spherically symmetric probability measures, in Geometric Aspects of Functional Analysis, Israel Seminar 2000–2001. Lecture Notes in Mathematics, vol. 1807 (Springer, Berlin, 2003), pp. 37–43Google Scholar
  9. 9.
    S.G. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton-Jacobi equations. J. Math. Pure Appl. 80(7), 669–696 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    F. Bolley, I. Gentil, A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equation. J. Funct. Anal. 263(8), 2430–2457 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    F. Bolley, I. Gentil, A. Guillin, Uniform convergence to equilibrium for granular media. Arch. Ration. Mech. Anal. 208(2), 429–445 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    F. Bolley, A. Guillin, F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation. Math. Model. Numer. Anal. 44(5), 867–884 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    C. Borell, Diffusion equations and geometric inequalities. Potential Anal. 12, 49–71 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    P. Cattiaux, A pathwise approach of some classical inequalities. Potential Anal. 20, 361–394 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    P. Cattiaux, Hypercontractivity for perturbed diffusion semi-groups. Ann. Fac. des Sc. de Toulouse 14(4), 609–628 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    P. Cattiaux, I. Gentil, A. Guillin, Weak logarithmic-Sobolev inequalities and entropic convergence. Probab. Theory Relat. Fields 139, 563–603 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    P. Cattiaux, A. Guillin, On quadratic transportation cost inequalities. J. Math. Pures Appl. 88(4), 341–361 (2006)MathSciNetGoogle Scholar
  18. 18.
    P. Cattiaux, A. Guillin, F. Malrieu, Probabilistic approach for granular media equations in the non uniformly convex case. Probab. Theory Relat. Fields 140, 19–40 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    P. Cattiaux, A. Guillin, P.A. Zitt, Poincaré inequalities and hitting times. Ann. Inst. Henri Poincaré. Probab. Stat. 49(1), 95–118 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    P. Cattiaux, C. Léonard, Minimization of the Kullback information of diffusion processes. Ann. Inst. Henri Poincaré. Prob. Stat. 30(1), 83–132 (1994); and correction in Ann. Inst. Henri Poincaré 31, 705–707 (1995)Google Scholar
  21. 21.
    J.F. Collet, F. Malrieu, Logarithmic Sobolev inequalities for inhomogeneous semigroups. ESAIM Probab. Stat. 12, 492–504 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    H. Djellout, A. Guillin, L. Wu, Transportation cost information inequalities for random dynamical systems and diffusions. Ann. Probab. 334, 1025–1028 (2002)Google Scholar
  23. 23.
    A. Eberle, Reflection coupling and Wasserstein contractivity without convexity. C. R. Acad. Sci. Paris Ser. I 349, 1101–1104 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    A. Eberle, Couplings, distances and contractivity for diffusion processes revisited. Available on Math. arXiv:1305.1233 [math.PR] (2013)Google Scholar
  25. 25.
    J. Fontbona, B. Jourdain, A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations. Available on Math. arXiv:1107.3300 [math.PR] (2011)Google Scholar
  26. 26.
    N. Gozlan, C. Léonard, Transport inequalities—a survey. Markov Process. Relat. Fields 16, 635–736 (2010)zbMATHGoogle Scholar
  27. 27.
    A. Guillin, F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality. J. Differ. Equ. 253(1), 20–40 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    A. Guillin, C. Léonard, L. Wu, N. Yao, Transportation-information inequalities for Markov processes. Probab. Theory Relat. Fields 144(3–4), 669–695 (2009)CrossRefzbMATHGoogle Scholar
  29. 29.
    N. Huet, Isoperimetry for spherically symmetric log-concave probability measures. Rev. Mat. Iberoam. 27(1), 93–122 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    N. Ikeda, S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd edn. (North-Holland, Amsterdam, 1988)Google Scholar
  31. 31.
    A.V. Kolesnikov, On diffusion semigroups preserving the log-concavity. J. Funct. Anal. 186(1), 196–205 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  32. 32.
    M. Ledoux, Logarithmic Sobolev inequalities for unbounded spin systems revisited, in Séminaire de Probabilités XXXV. Lecture Notes in Mathematics, vol. 1755 (Springer, New York, 2001), pp. 167–194Google Scholar
  33. 33.
    J. Lehec, Representation formula for the entropy and functional inequalities. Ann. Inst. Henri Poincaré. Prob. Stat. 49(3), 885–899 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    T. Lindvall, L.C.G. Rogers, Coupling of multidimensional diffusions by reflection. Ann. Probab. 14, 860–872 (1986)CrossRefMathSciNetzbMATHGoogle Scholar
  35. 35.
    F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stoch. Process. Appl. 95(1), 109–132 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    F. Otto, C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  37. 37.
    J. Pitman, M. Yor, Bessel processes and infinitely divisible laws, in Stochastic Integrals. Lecture Notes in Mathematics, vol. 851 (Springer, New York, 1980), pp. 285–370Google Scholar
  38. 38.
    P.E. Protter, Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability, vol. 21 (Springer, Berlin, 2005)Google Scholar
  39. 39.
    M. Röckner, F.Y. Wang, Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Funct. Anal. 185(2), 564–603 (2001)CrossRefMathSciNetzbMATHGoogle Scholar
  40. 40.
    M. Röckner, F.Y. Wang, Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. Anal. Quant. Probab. Relat. Top. 13, 27–37 (2010)CrossRefzbMATHGoogle Scholar
  41. 41.
    C. Villani, Hypocoercivity. Mem. Am. Math. Soc. 202(950), iv+141 (2009)Google Scholar
  42. 42.
    C. Villani, Optimal Transport. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338 (Springer, Berlin, 2009)Google Scholar
  43. 43.
    M. Von Renesse, K.T. Sturm, Transport inequalities, gradient estimates, entropy, and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)CrossRefzbMATHGoogle Scholar
  44. 44.
    F.Y. Wang, Functional Inequalities, Markov Processes and Spectral Theory (Science Press, Beijing, 2004)Google Scholar
  45. 45.
    F.Y. Wang, T. Zhang, Log-Harnack inequality for mild solutions of SPDEs with strongly multiplicative noise. Available on Math. arXiv:1210.6416 [math.PR] (2012)Google Scholar
  46. 46.
    C. Yi, On the first passage time distribution of an Ornstein-Uhlenbeck process. Quant. Fin. 10(9), 957–960 (2010)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Institut de Mathématiques de ToulouseUniversité de Toulouse, CNRS UMR 5219Toulouse cedex 09France
  2. 2.Laboratoire de Mathématiques, CNRS UMR 6620Université Blaise PascalAubièreFrance

Personalised recommendations