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

Abstract

Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions 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 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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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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Keywords

  • Generalized gradient flows
  • Large deviations
  • Convex analysis
  • Particle systems

Mathematics Subject Classifications (2010)

  • 35Q82
  • 35Q84
  • 49S05
  • 60F10
  • 60J25
  • 60J27