Journal of Statistical Physics

, Volume 152, Issue 2, pp 237–274 | Cite as

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

  • T. Lelièvre
  • F. Nier
  • G. A. Pavliotis


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.


Non-reversible diffusion Convergence to equilibrium Wick calculus 



We would like to thank Matthieu Dubois for preliminary numerical experiments.


  1. 1.
    Ammari, Z., Nier, F.: Mean field limit for bosons and infinite dimensional phase-space analysis. Ann. Henri Poincaré 9, 1503–1574 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  2. 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) MathSciNetGoogle Scholar
  3. 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) MathSciNetGoogle Scholar
  4. 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) MathSciNetADSMATHCrossRefGoogle Scholar
  5. 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) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Constantin, P., Kiselev, A., Ryzhik, L., Zlatos, A.: Diffusion and mixing in fluid flow. Ann. Math. 168(2), 643–674 (2008) MathSciNetMATHCrossRefGoogle Scholar
  7. 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) CrossRefGoogle Scholar
  8. 8.
    Dencker, N., Sjöstrand, J., Zworski, M.: Pseudospectra of semi-classical (pseudo)differential operator. Commun. Pure Appl. Math. 57(3), 384–415 (2004) MATHCrossRefGoogle Scholar
  9. 9.
    Diaconis, P.: The Markov chain Monte Carlo revolution. Bull. Am. Math. Soc. 46(2), 179–205 (2009) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Diaconis, P., Miclo, L.: On the spectral analysis of second-order Markov chains (2012).
  11. 11.
    Eckmann, J.P., Hairer, M.: Spectral properties of hypoelliptic operators. Commun. Math. Phys. 235, 233–253 (2003) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Engel, K., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equation. Graduate Texts in Mathematics, vol. 194. Springer, Berlin (2000) Google Scholar
  13. 13.
    Fontbona, J., Jourdain, B.: A trajectorial interpretation of the dissipations of entropy and Fisher information for stochastic differential equations (2011).
  14. 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) MathSciNetMATHCrossRefGoogle Scholar
  15. 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) MathSciNetGoogle Scholar
  16. 16.
    Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. B 73(2), 1–37 (2011) MathSciNetGoogle Scholar
  17. 17.
    Helffer, B.: Théorie spectrale pour des opérateurs globalement elliptiques Astérisque 112 (1984) Google Scholar
  18. 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) MATHGoogle Scholar
  19. 19.
    Helffer, B., Sjöstrand, J.: From resolvent bounds to semigroup bounds. In: Proceedings of the Meeting Equations aux Dérivées Partielles, Evian (2009) Google Scholar
  20. 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) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hitrik, M., Pravda-Starov, K.: Spectra and semigroup smoothing for non-elliptic quadratic operators. Math. Ann. 344(4), 801–846 (2009) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Hörmander, L.: Symplectic classification of quadratic forms, and general Mehler formulas. Math. Z. 219(3), 413–449 (1995) MathSciNetMATHCrossRefGoogle Scholar
  23. 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 MATHGoogle Scholar
  24. 24.
    Hwang, C.R., Hwang-Ma, S.Y., Sheu, S.J.: Accelerating Gaussian diffusions. Ann. Appl. Probab. 3(3), 897–913 (1993) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Hwang, C.R., Hwang-Ma, S.Y., Sheu, S.J.: Accelerating diffusions. Ann. Appl. Probab. 15(2), 1433–1444 (2005) MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Lelièvre, T.: Two mathematical tools to analyze metastable stochastic processes (2012). arXiv:1201.3775
  27. 27.
    Lelièvre, T., Rousset, M., Stoltz, G.: Free Energy Computations: A Mathematical Perspective. Imperial College Press, London (2010) MATHCrossRefGoogle Scholar
  28. 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) MATHCrossRefGoogle Scholar
  29. 29.
    Lorenzi, L., Bertoldi, M.: Analytical Methods for Markov Semigroups. CRC Press, New York (2006) CrossRefGoogle Scholar
  30. 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) MathSciNetMATHGoogle Scholar
  31. 31.
    Metafune, G., Pallara, D., Priola, E.: Spectrum of Ornstein-Uhlenbeck operators in L p spaces with respect to invariant measures. J. Funct. Anal. 196(1), 40–60 (2002) MathSciNetMATHCrossRefGoogle Scholar
  32. 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) CrossRefGoogle Scholar
  33. 33.
    Øksendal, B.: Stochastic Differential Equations. Universitext. Springer, Berlin (2003) CrossRefGoogle Scholar
  34. 34.
    Ottobre, M., Pavliotis, G.A., Pravda-Starov, K.: Exponential return to equilibrium for hypoelliptic Ornstein-Uhlenbeck processes (2012, in preparation) Google Scholar
  35. 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) MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Pravda-Starov, K.: Contraction semigroups of elliptic quadratic differential operators. Math. Z. 259(2), 363–391 (2008) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Pravda-Starov, K.: On the pseudospectrum of elliptic quadratic differential operators. Duke Math. J. 145(2), 249–279 (2008) MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1975) MATHGoogle Scholar
  39. 39.
    Sjöstrand, J.: Parametrices for pseudodifferential operators with multiple characteristics. Ark. Mat. 12, 85–130 (1974) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Snyders, J., Zakai, M.: On nonnegative solutions of the equation AD+DA′=−C. SIAM J. Appl. Math. 18(3), 704–714 (1970) MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Trefethen, L., Embree, M.: Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators. Princeton University Press, Princeton (2005) Google Scholar
  42. 42.
    Villani, C.: Hypocoercivity. Memoirs Amer. Math. Soc. 202 (2009) Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CERMICS, Ecole des pontsUniversité Paris-EstMarne la Vallée cedex 2France
  2. 2.MicMac project teamINRIALe Chesnay cedexFrance
  3. 3.IRMAR, Campus de BeaulieuUniversité de Rennes 1RennesFrance
  4. 4.Department of MathematicsImperial College LondonLondonEngland

Personalised recommendations