Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion
详细信息 查看全文
- 作者：T. Lelièvre (1) (2)
F. Nier (3)
G. A. Pavliotis (4) - 关键词：Non ; reversible diffusion ; Convergence to equilibrium ; Wick calculus
- 刊名：Journal of Statistical Physics
- 出版年：2013
- 出版时间：July 2013
- 年：2013
- 卷：152
- 期：2
- 页码：237-274
- 全文大小：1377KB
- 参考文献：1. Ammari, Z., Nier, F.: Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré
2. Ammari, Z., Nier, F.: Mean field propagation of Wigner measures and general BBGKY hierarchies for general bosonic states. J. Math. Pures Appl.
3. Arnold, A., Carlen, E., Ju, Q.: Large-time behavior of non-symmetric Fokker-Planck type equations. Commun. Stoch. Anal.
4. Berestycki, H., Hamel, F., Nadirashvili, N.: Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Commun. Math. Phys.
5. Chopin, N., Lelièvre, T., Stoltz, G.: Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors. Stat. Comput.
6. Constantin, P., Kiselev, A., Ryzhik, L., Zlatos, A.: Diffusion and mixing in fluid flow. Ann. Math.
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. 141-51. Amer. Math. Soc., Providence (2007) g/10.1090/pspum/076.1/2310201">CrossRef
8. Dencker, N., Sj?strand, J., Zworski, M.: Pseudospectra of semi-classical (pseudo)differential operator. Commun. Pure Appl. Math.
9. Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc.
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.
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). g/abs/1107.3300" class="a-plus-plus">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.
15. Gallagher, I., Gallay, T., Nier, F.: Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator. Int. Math. Res. Not.
16. Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J.?R.?Stat. Soc. B
17. Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques Astérisque
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., Sj?strand, 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.
21. Hitrik, M., Pravda-Starov, K.: Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann.
22. H?rmander, L.: Symplectic classification of quadratic forms, and general Mehler formulas. Math. Z.
23. H?rmander, 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.
25. Hwang, C.R., Hwang-Ma, S.Y., Sheu, S.J.: Accelerating diffusions. Ann. Appl. Probab.
26. Lelièvre, T.: Two mathematical tools to analyze metastable stochastic processes (2012). g/abs/arXiv:1201.3775" class="a-plus-plus">arXiv:1201.3775
27. Lelièvre, T., Rousset, M., Stoltz, G.: Free Energy Computations: A?Mathematical Perspective. Imperial College Press, London (2010) g/10.1142/9781848162488">CrossRef
28. Lerner, N.: Metrics on the Phase Space and Non-selfadjoint Pseudo-differential Operators. Pseudo-differential Operators. Theory and Applications, vol. 3. Birkh?user, Basel (2010) g/10.1007/978-3-7643-8510-1">CrossRef
29. Lorenzi, L., Bertoldi, M.: Analytical Methods for Markov Semigroups. CRC Press, New York (2006) g/10.1201/9781420011586">CrossRef
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.
31. Metafune, G., Pallara, D., Priola, E.: Spectrum of Ornstein-Uhlenbeck operators in / L / p spaces with respect to invariant measures. J. Funct. Anal.
32. Metzner, P., Schütte, C., Vanden-Eijnden, E.: Illustration of transition path theory on a collection of simple examples. J. Chem. Phys.
33. ?ksendal, B.: Stochastic Differential Equations. Universitext. Springer, Berlin (2003) g/10.1007/978-3-642-14394-6">CrossRef
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.
36. Pravda-Starov, K.: Contraction semigroups of elliptic quadratic differential operators. Math. Z.
37. Pravda-Starov, K.: On the pseudospectrum of elliptic quadratic differential operators. Duke Math. J.
38. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1975)
39. Sj?strand, J.: Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat.
40. Snyders, J., Zakai, M.: On nonnegative solutions of the equation / AD+ / DA-?em class="a-plus-plus">C. SIAM J. Appl. Math.
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. - 作者单位：T. Lelièvre (1) (2)
F. Nier (3)
G. A. Pavliotis (4)
1. CERMICS, Ecole des ponts, Université Paris-Est, 6-8 avenue Blaise Pascal, 77455, Marne la Vallée cedex 2, France
2. MicMac project team, INRIA, 78153, Le Chesnay cedex, France
3. IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042, Rennes, France
4. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, England
文摘
We consider non-reversible perturbations of reversible diffusions that do not alter the invariant distribution and we ask whether there exists an optimal perturbation such that the rate of convergence to equilibrium is maximized. We solve this problem for the case of linear drift by proving the existence of such optimal perturbations and by providing an easily implementable algorithm for constructing them. We discuss in particular the role of the prefactor in the exponential convergence estimate. Our rigorous results are illustrated by numerical experiments.