On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion

A. Mielke, D.R.M. Renger, M.A. Peletier

Research output: Contribution to journalArticleAcademicpeer-review

53 Citations (Scopus)
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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 L(¿t,¿¿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. Keywords: Generalized gradient flows · Large deviations · Convex analysis · Particle systems
Original languageEnglish
Pages (from-to)1293-1327
Number of pages35
JournalPotential Analysis
Volume41
Issue number4
DOIs
Publication statusPublished - 2014

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