Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion

Journal of Statistical Physics

Volumn 152, Issue 2, 2013, Pages 237-274

  Lelievre T ^a,b   Nier, F ^c   Pavliotis, G A ^d  

^a UNIVERSITÉ PARIS EST   (France)

^b INRIA   (France)

^c UNIVERSITÉ DE RENNES 1   (France)

^d IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Convergence to equilibrium; Non reversible diffusion; Wick calculus

Indexed keywords

EID: 84880840823     PISSN: 00224715     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10955-013-0769-x     Document Type: Article

References (42)

1. Ammari, Z., Nier, F.: Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré 9, 1503-1574 (2008).
2. Ammari, Z., Nier, F.: Mean field propagation of Wigner measures and general BBGKY hierarchies for general bosonic states. J. Math. Pures Appl. 95, 585-626 (2010).
3. Arnold, A., Carlen, E., Ju, Q.: Large-time behavior of non-symmetric Fokker-Planck type equations. Commun. Stoch. Anal. 2(1), 153-175 (2008).
4. Berestycki, H., Hamel, F., Nadirashvili, N.: Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Commun. Math. Phys. 253(2), 451-480 (2005).
5. Chopin, N., Lelièvre, T., Stoltz, G.: Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors. Stat. Comput. 22(4), 897-916 (2012).
6. Constantin, P., Kiselev, A., Ryzhik, L., Zlatos, A.: Diffusion and mixing in fluid flow. Ann. Math. 168(2), 643-674 (2008).
7. Davies, E. B.: Non-self-adjoint operators and pseudospectra. In: Spectral Theory and Mathematical Physics: a Festschrift in Honor of Barry Simon's 60th Birthday. Proc. Sympos. Pure Math., vol. 76, pp. 141-151. Amer. Math. Soc., Providence (2007).
8. Dencker, N., Sjöstrand, J., Zworski, M.: Pseudospectra of semi-classical (pseudo)differential operator. Commun. Pure Appl. Math. 57(3), 384-415 (2004).
9. Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc. 46(2), 179-205 (2009).
10. Diaconis, P., Miclo, L.: On the spectral analysis of second-order Markov chains (2012). http://hal. archives-ouvertes. fr/hal-00719047/.
11. Eckmann, J. P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235, 233-253 (2003).
12. Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equation. Graduate Texts in Mathematics, vol. 194. Springer, Berlin (2000).
13. Fontbona, J., Jourdain, B.: A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations (2011). http://arxiv. org/abs/1107. 3300.
14. Franke, B., Hwang, C. R., Pai, H. M., Sheu, S. J.: The behavior of the spectral gap under growing drift. Trans. Am. Math. Soc. 362(3), 1325-1350 (2010).
15. Gallagher, I., Gallay, T., Nier, F.: Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not. 12, 2147-2199 (2009).
16. Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. B 73(2), 1-37 (2011).
17. Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques Astérisque 112 (1984).
18. Helffer, B., Nier, F.: Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians. Lecture Notes in Mathematics, vol. 1862. Springer, Berlin (2005).
19. Helffer, B., Sjöstrand, J.: From resolvent bounds to semigroup bounds. In: Proceedings of the Meeting Equations aux Dérivées Partielles, Evian (2009).
20. Hérau, F., Nier, F.: Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Ration. Mech. Anal. 171(2), 151-218 (2004).
21. Hitrik, M., Pravda-Starov, K.: Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann. 344(4), 801-846 (2009).
22. Hörmander, L.: Symplectic classification of quadratic forms, and general Mehler formulas. Math. Z. 219(3), 413-449 (1995).
23. Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Classics in Mathematics. Springer, Berlin (2007). Pseudo-differential operators, Reprint of the 1994 edition.
24. Hwang, C. R., Hwang-Ma, S. Y., Sheu, S. J.: Accelerating Gaussian diffusions. Ann. Appl. Probab. 3(3), 897-913 (1993).
25. Hwang, C. R., Hwang-Ma, S. Y., Sheu, S. J.: Accelerating diffusions. Ann. Appl. Probab. 15(2), 1433-1444 (2005).
26. Lelièvre, T.: Two mathematical tools to analyze metastable stochastic processes (2012). arXiv: 1201. 3775.
27. Lelièvre, T., Rousset, M., Stoltz, G.: Free Energy Computations: A Mathematical Perspective. Imperial College Press, London (2010).
28. Lerner, N.: Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators. Pseudo-differential Operators. Theory and Applications, vol. 3. Birkhäuser, Basel (2010).
29. Lorenzi, L., Bertoldi, M.: Analytical Methods for Markov Semigroups. CRC Press, New York (2006).
30. Markowich, P. A., Villani, C.: On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis. Mat. Contemp. 19, 1-29 (2000).
31. p spaces with respect to invariant measures. J. Funct. Anal. 196(1), 40-60 (2002).
32. Metzner, P., Schütte, C., Vanden-Eijnden, E.: Illustration of transition path theory on a collection of simple examples. J. Chem. Phys. 125(8), 084, 110 (2006).
33. Øksendal, B.: Stochastic Differential Equations. Universitext. Springer, Berlin (2003).
34. Ottobre, M., Pavliotis, G. A., Pravda-Starov, K.: Exponential return to equilibrium for hypoelliptic Ornstein-Uhlenbeck processes (2012, in preparation).
35. Ottobre, M., Pavliotis, G. A., Pravda-Starov, K.: Exponential return to equilibrium for hypoelliptic quadratic systems. J. Funct. Anal. 262(9), 4000-4039 (2012).
36. Pravda-Starov, K.: Contraction semigroups of elliptic quadratic differential operators. Math. Z. 259(2), 363-391 (2008).
37. Pravda-Starov, K.: On the pseudospectrum of elliptic quadratic differential operators. Duke Math. J. 145(2), 249-279 (2008).
38. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1975).
39. Sjöstrand, J.: Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat. 12, 85-130 (1974).
40. Snyders, J., Zakai, M.: On nonnegative solutions of the equation AD+DA′=-C. SIAM J. Appl. Math. 18(3), 704-714 (1970).
41. Trefethen, L., Embree, M.: Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005).
42. Villani, C.: Hypocoercivity. Memoirs Amer. Math. Soc. 202 (2009).