Skip to main content
SpringerLink
Log in

Couplings, gradient estimates and logarithmic Sobolev inequalitiy for Langevin bridges

  • Published:
Probability Theory and Related Fields Aims and scope Submit manuscript

Abstract

In this paper we establish quantitative results about the bridges of the Langevin dynamics and the associated reciprocal processes. They include an equivalence between gradient estimates for bridge semigroups and couplings, comparison principles, bounds of the distance between bridges of different Langevin dynamics, and a logarithmic Sobolev inequality for bridge measures. The existence of an invariant measure for the bridges is also discussed and quantitative bounds for the convergence to the invariant measure are proven. All results are based on a seemingly new expression of the drift of a bridge in terms of the reciprocal characteristic, which, roughly speaking, quantifies the “mean acceleration” of a bridge.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, vol. 348. Springer, Berlin (2013)

    MATH  Google Scholar 

  2. Barbour, A.D.: Stein’s method for diffusion approximations. Probab. Theory Relat. Fields 84(3), 297–322 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  3. Benjamini, I., Lee, S.: Conditioned diffusions which are Brownian bridges. J. Theor. Probab. 10(3), 733–736 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernstein, S.: Sur les liaisons entre les grandeurs aléatoires. In: Vehr. des intern. Mathematikerkongress Zürich, vol. I. (1932)

  5. Brascamp, H.J., Lieb, E.H.: On extensions of the Brunn-Minkowski and Prékopa-leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)

    Article  MATH  Google Scholar 

  6. Capitaine, M., Hsu, E.P., Ledoux, M., et al.: Martingale representation and a simple proof of logarithmic sobolev inequalities on path spaces. Electron. Commun. Probab. 2, 71–81 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chiarini, A., Cipriani, A., Conforti, G.: Approximating conditional distributions (2017). arXiv:1710.08856

  8. Clark, J.M.C.: A local characterization of reciprocal diffusions. Appl. Stoch. Anal. 5, 45–59 (1991)

    MathSciNet  MATH  Google Scholar 

  9. Conforti, G.: Coupling bridges on riemannian manifolds (in preparation)

  10. Conforti, G.: Fluctuations of bridges, reciprocal characteristics, and concentration of measure. arXiv:1602.07231, to appear in Annales de l’Institut Henri Poincaré (2016)

  11. Cruzeiro, A.B., Zambrini, J.C.: Malliavin calculus and Euclidean quantum mechanics. I. Functional calculus. J. Funct. Anal. 96(1), 62–95 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  12. Doob, J.L.: Conditional Brownian motion and the boundary limits of harmonic functions. Bulletin de la Socit Mathmatique de France 85, 431–458 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  13. Doss, H., Royer, G.: Processus de diffusion associe aux mesures de gibbs sur. Probab. Theory Relat. Fields 46(1), 107–124 (1978)

    MathSciNet  MATH  Google Scholar 

  14. Driver, B.K., Lohrenz, T.: Logarithmic sobolev inequalities for pinned loop groups. J. Funct. Anal. 140(2), 381–448 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Föllmer, H.: Random fields and diffusion processes. In: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–1987, pp. 101–203. Springer (1988)

  16. Gross, L.: Logarithmic sobolev inequalities on loop groups. J. Funct. Anal. 102(2), 268–313 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hairer, M., Stuart, A.M., Voss, J., et al.: Analysis of spdes arising in path sampling part ii: the nonlinear case. Ann. Appl. Probab. 17(5/6), 1657–1706 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hairer, M., Stuart, A.M., Voss, J., Wiberg, P., et al.: Analysis of spdes arising in path sampling. Part i: the gaussian case. Commun. Math. Sci. 3(4), 587–603 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hsu, E.P.: Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Commun. Math. Phys. 189(1), 9–16 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jamison, B.: Reciprocal processes: the stationary Gaussian case. Ann. Math. Stat. 41(5), 1624–1630 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jamison, B.: Reciprocal processes. Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete 30(1), 65–86 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jamison, B.: The Markov processes of Schrdinger. Zeitschrift fr Wahrscheinlichkeitstheorie und Verwandte Gebiete 32(4), 323–331 (1975)

    Article  MATH  Google Scholar 

  23. Krener, A.J.: Reciprocal diffusions and stochastic differential equations of second order. Stochastics 107(4), 393–422 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  24. Krener, A.J.: Reciprocal diffusions in flat space. Probab. Theory Relat. Fields 107(2), 243–281 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ledoux, M.: The Concentration of Measure Phenomenon Volume 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2001)

  26. Léonard, C.: A survey of the Schrdinger problem and some of its connections with optimal transport. Discrete Contin. Dyn. Syst. 34(4), 1533–1574 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Léonard, C., Rœlly, S., Zambrini, J.C.: Reciprocal processes. A measure-theoretical point of view. Probab. Surv. 11, 237–269 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  28. Levy, B.C., Krener, A.J.: Dynamics and kinematics of reciprocal diffusions. J. Math. Phys. 34(5), 1846–1875 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lörinczi, J., Hiroshima, F., Betz, V.: Feynman-Kac-type Theorems and Gibbs Measures on Path Space: With Applications to Rigorous Quantum Field Theory, vol. 34. Walter de Gruyter, Berlin (2011)

    Book  MATH  Google Scholar 

  30. Nelson, E.: Dynamical Theories of Brownian Motion, vol. 2. Princeton University Press, Princeton (1967)

    MATH  Google Scholar 

  31. Priouret, P., Yor, M.: Processus de diffusion à valeurs dans R et mesures quasi-invariantes sur C (R, R). Societe Mathematique de France, Marseille (1975)

    MATH  Google Scholar 

  32. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, vol. 293. Springer, Berlin (2013)

    MATH  Google Scholar 

  33. Reznikoff, M.G., Vanden-Eijnden, E.: Invariant measures of stochastic partial differential equations and conditioned diffusions. C.R. Math. 340(4), 305–308 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rœlly, S., Thieullen, M.: A characterization of reciprocal processes via an integration by parts formula on the path space. Probab. Theory Relat. Fields 123(1), 97–120 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Rœlly, S., Thieullen, M.: Duality formula for the bridges of a brownian diffusion: application to gradient drifts. Stoch. Process. Appl. 115(10), 1677–1700 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. Royer, G.: An Initiation to Logarithmic Sobolev Inequalities, vol. 5. American Mathematical Society, Providence (2007)

    MATH  Google Scholar 

  37. Roynette, B., Yor, M.: Penalising Brownian Paths, vol. 1969. Springer, Berlin (2009)

    MATH  Google Scholar 

  38. Schrödinger, E.: Über die Umkehrung der Naturgesetze. Sitzungsberichte Preuss. Akad. Wiss. Berlin. Phys. Math. 144, 144–153 (1931)

  39. Schrödinger, E.: La théorie relativiste de l’électron et l’ interprétation de la mécanique quantique. Ann. Inst Henri Poincaré 2, 269–310 (1932)

    MathSciNet  MATH  Google Scholar 

  40. Simon, B.: Convexity: An Analytic Viewpoint, vol. 187. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  41. Thieullen, M.: Second order stochastic differential equations and non-Gaussian reciprocal diffusions. Probab. Theory Relat. Fields 97(1–2), 231–257 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  42. Thieullen, M., Zambrini, J.C.: Symmetries in the stochastic calculus of variations. Probab. Theory Relat. Fields 107(3), 401–427 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  43. von Renesse, M.-K., Sturm, K.-T.: Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. 58(7), 923–940 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. Zambrini, J.C.: Variational processes and stochastic versions of mechanics. J. Math. Phys. 27(9), 2307–2330 (1986)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Centre des Mathématiques Appliquées, École Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France

    Giovanni Conforti

  2. Fakultät für Mathematik und Informatik, Universität Leipzig, Augustusplatz 10, 04109, Leipzig, Germany

    Max Von Renesse

Authors
  1. Giovanni Conforti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Max Von Renesse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Giovanni Conforti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Conforti, G., Von Renesse, M. Couplings, gradient estimates and logarithmic Sobolev inequalitiy for Langevin bridges. Probab. Theory Relat. Fields 172, 493–524 (2018). https://doi.org/10.1007/s00440-017-0814-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-017-0814-9

Keywords

Mathematics Subject Classification

Search

Navigation