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Convergence rates for nonequilibrium Langevin dynamics

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Abstract

We study the exponential convergence to the stationary state for nonequilibrium Langevin dynamics, by a perturbative approach based on hypocoercive techniques developed for equilibrium Langevin dynamics. The Hamiltonian and overdamped limits (corresponding respectively to frictions going to zero or infinity) are carefully investigated. In particular, the maximal magnitude of admissible perturbations are quantified as a function of the friction. Numerical results based on a Galerkin discretization of the generator of the dynamics confirm the theoretical lower bounds on the spectral gap.

Résumé

Nous considérons la convergence exponentielle vers l’état stationnaire pour des dynamiques de Langevin hors d’équilibre, par une approche perturbative reposant sur des techniques d’hypocoercivité initialement développées pour des dynamiques d’équilibre. Les limites hamiltoniennes et suramorties (qui correspondent respectivement au cas des frictions tendant vers zéro ou l’infini) sont étudiées précisément. En particulier, nous quantifions la magnitude maximale des perturbations admissibles en fonction de la friction. Des simulations numériques utilisant une discrétisation de Galerkin du générateur de la dynamique confirment les bornes inférieures que nous obtenons théoriquement pour le trou spectral.

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Notes

  1. Although sharper results may be obtained with Young inequalities, the final scaling of admissible values of \(\tau \) in terms of \(\xi \) is unaffected.

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Acknowledgements

We thank Anton Arnold for interesting discussions. The work of G.S. is supported by the Agence Nationale de la Recherche, under Grant ANR-14-CE23-0012 (COSMOS) and by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492. The work of S.O. is supported by the Agence Nationale de la Recherche, under Grant ANR-15-CE40-0020-01 (LSD).

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Authors and Affiliations

  1. CEREMADE, UMR-CNRS, Université Paris Dauphine, PSL, Place du Maréchal De Lattre De Tassigny, 75775, Paris Cedex 16, France

    A. Iacobucci & S. Olla

  2. Université Paris-Est, CERMICS (ENPC), Inria, 77455, Marne-la-Vallée, France

    G. Stoltz

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  1. A. Iacobucci
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  2. S. Olla
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  3. G. Stoltz
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Correspondence to G. Stoltz.

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Iacobucci, A., Olla, S. & Stoltz, G. Convergence rates for nonequilibrium Langevin dynamics. Ann. Math. Québec 43, 73–98 (2019). https://doi.org/10.1007/s40316-017-0091-0

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  • DOI: https://doi.org/10.1007/s40316-017-0091-0

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