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On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion

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Abstract

Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions ℒ that induce a flow, given by \(\mathcal{L} (\rho _{t},\dot \rho _{t})=0\). We derive necessary and sufficient conditions for the unique existence of a generalized gradient structure for the induced flow, as well as explicit formulas for the corresponding driving entropy and dissipation functional. In particular, we show how these conditions can be given a probabilistic interpretation when ℒ is associated to the large deviations of a microscopic particle system. Finally, we illustrate the theory for independent Brownian particles with drift, which leads to the entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.

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Authors and Affiliations

  1. Weierstrass-Institut für, Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117, Berlin, Germany

    A. Mielke & D. R. M. Renger

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Rudower Chaussee 25, Adlershof, 12489, Berlin, Germany

    A. Mielke

  3. Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600, MB, Eindhoven, the Netherlands

    M. A. Peletier

  4. Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology, P.O. Box 513, 5600, MB, Eindhoven, the Netherlands

    M. A. Peletier

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  1. A. Mielke
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  2. M. A. Peletier
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  3. D. R. M. Renger
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Correspondence to D. R. M. Renger.

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Mielke, A., Peletier, M.A. & Renger, D.R.M. On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion. Potential Anal 41, 1293–1327 (2014). https://doi.org/10.1007/s11118-014-9418-5

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  • DOI: https://doi.org/10.1007/s11118-014-9418-5

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