The Annals of Probability

A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations

Joaquin Fontbona and Benjamin Jourdain

Full-text: Open access

Abstract

The dissipation of general convex entropies for continuous time Markov processes can be described in terms of backward martingales with respect to the tail filtration. The relative entropy is the expected value of a backward submartingale. In the case of (not necessarily reversible) Markov diffusion processes, we use Girsanov theory to explicit the Doob–Meyer decomposition of this submartingale. We deduce a stochastic analogue of the well-known entropy dissipation formula, which is valid for general convex entropies, including the total variation distance. Under additional regularity assumptions, and using Itô’s calculus and ideas of Arnold, Carlen and Ju, we obtain moreover a new Bakry–Emery criterion which ensures exponential convergence of the entropy to $0$. This criterion is nonintrinsic since it depends on the square root of the diffusion matrix and cannot be written only in terms of the diffusion matrix itself. We provide examples where the classic Bakry–Emery criterion fails, but our nonintrinsic criterion applies without modifying the law of the diffusion process.

Article information

Source
Ann. Probab., Volume 44, Number 1 (2016), 131-170.

Dates
Received: July 2013
Revised: February 2014
First available in Project Euclid: 2 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1454423037

Digital Object Identifier
doi:10.1214/14-AOP969

Mathematical Reviews number (MathSciNet)
MR3456334

Zentralblatt MATH identifier
1351.60070

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60H30: Applications of stochastic analysis (to PDE, etc.) 37A35: Entropy and other invariants, isomorphism, classification 26D10: Inequalities involving derivatives and differential and integral operators 35B40: Asymptotic behavior of solutions

Keywords
Long-time behavior stochastic differential equations Bakry–Emery criterion convex Sobolev inequalities time reversal Girsanov theory

Citation

Fontbona, Joaquin; Jourdain, Benjamin. A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations. Ann. Probab. 44 (2016), no. 1, 131--170. doi:10.1214/14-AOP969. https://projecteuclid.org/euclid.aop/1454423037


Export citation

References

  • [1] Arnold, A., Carlen, E. and Ju, Q. (2008). Large-time behavior of non-symmetric Fokker–Planck type equations. Commun. Stoch. Anal. 2 153–175.
  • [2] Arnold, A., Markowich, P., Toscani, G. and Unterreiter, A. (2001). On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Comm. Partial Differential Equations 26 43–100.
  • [3] Bakry, D. and Émery, M. (1985). Diffusions hypercontractives. In Séminaire de Probabilités, XIX, 1983/1984 177–206. Springer, Berlin.
  • [4] Bouleau, N. and Hirsch, F. (1991). Dirichlet Forms and Analysis on Wiener Space. De Gruyter Studies in Mathematics 14. de Gruyter, Berlin.
  • [5] Cattiaux, P. (2004). A pathwise approach of some classical inequalities. Potential Anal. 20 361–394.
  • [6] Cattiaux, P. and Léonard, C. (1994). Minimization of the Kullback information of diffusion processes. Ann. Inst. Henri Poincaré Probab. Stat. 30 83–132.
  • [7] Chafaï, D. (2004). Entropies, convexity, and functional inequalities: On $\Phi$-entropies and $\Phi$-Sobolev inequalities. J. Math. Kyoto Univ. 44 325–363.
  • [8] Föllmer, H. (1986). Time reversal on Wiener space. In Stochastic Processes—Mathematics and Physics (Bielefeld, 1984). Lecture Notes in Math. 1158 119–129. Springer, Berlin.
  • [9] Föllmer, H. and Wakolbinger, A. (1986). Time reversal of infinite-dimensional diffusions. Stochastic Process. Appl. 22 59–77.
  • [10] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs, NJ.
  • [11] Haussmann, U. G. and Pardoux, É. (1986). Time reversal of diffusions. Ann. Probab. 14 1188–1205.
  • [12] Holley, R. and Stroock, D. (1987). Logarithmic Sobolev inequalities and stochastic Ising models. J. Stat. Phys. 46 1159–1194.
  • [13] Hwang, C.-R., Hwang-Ma, S.-Y. and Sheu, S.-J. (2005). Accelerating diffusions. Ann. Appl. Probab. 15 1433–1444.
  • [14] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [15] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [16] Kusuoka, S. and Stroock, D. (1985). Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 1–76.
  • [17] Millet, A., Nualart, D. and Sanz, M. (1989). Integration by parts and time reversal for diffusion processes. Ann. Probab. 17 208–238.
  • [18] Pardoux, É. (1986). Grossissement d’une filtration et retournement du temps d’une diffusion. In Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math. 1204 48–55. Springer, Berlin.