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

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Abstract

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.

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Acknowledgement

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

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Authors and Affiliations

  1. CERMICS, Ecole des ponts, Université Paris-Est, 6-8 avenue Blaise Pascal, 77455, Marne la Vallée cedex 2, France

    T. Lelièvre

  2. MicMac project team, INRIA, 78153, Le Chesnay cedex, France

    T. Lelièvre

  3. IRMAR, Campus de Beaulieu, Université de Rennes 1, 35042, Rennes, France

    F. Nier

  4. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, England

    G. A. Pavliotis

Authors
  1. T. Lelièvre
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  2. F. Nier
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  3. G. A. Pavliotis
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Corresponding author

Correspondence to T. Lelièvre.

Additional information

This work is supported by the Agence Nationale de la Recherche, under grant ANR-09-BLAN-0216-01 (MEGAS). This work was initiated while the two last authors had a INRIA-sabbatical semester and year in the INRIA-project team MICMAC. The research of GP is partially supported by the EPSRC under grants No. EP/H034587 and No. EP/J009636/1.

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Lelièvre, T., Nier, F. & Pavliotis, G.A. Optimal Non-reversible Linear Drift for the Convergence to Equilibrium of a Diffusion. J Stat Phys 152, 237–274 (2013). https://doi.org/10.1007/s10955-013-0769-x

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