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

By Alexander Mielke, Mark A. Peletier, Michiel Renger. Potential Analysis, 2013

194 citations recorded by the source corpus.

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(ρt,ρ̇t)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal{L} (\rho _{t},\dot \rho _{t})=0$\end{document}. 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.

Subfields: Applied Mathematics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/19027aa95f66f02c6a166472ee9115bd8ec428b6

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In Applied Mathematics

In Statistical and Nonlinear Physics

Citations in the corpus

Jump processes as generalized gradient flows (2020, methodology): “Both of these are based not on the quadratic energy φ(s) = 1 2 s(2), but on the Boltzmann entropy functional φ(s) = s log s− s+ 1: (1) The large-deviation characterization [MPR14] leads to the choice Ψ(ξ) := 4 ( cosh(ξ/2)− 1 ) and α(u, v) := √ uv.”