2013/12/29 by Alexander Mielke, A. Mielke, D. R. Michiel Renger +3 · 1 citation
Mathematics · Physics and Astronomy · #math-ph #math.AP #math.FA #math.MP #math.PR #msc:35Q82 #msc:35Q84 #msc:49S05 #msc:60F10 #msc:60J25 #msc:60J27
paper · pdf · doi:10.1007/s11118-014-9418-5
published as Potential Analysis 41:4 (2014) 1293-1327
arxiv created 2013/12/29 · crossref issued 2014/06/15 · crossref published 2014/06/15 · crossref published-online 2014/06/15 · crossref created 2014/06/19 · crossref published-print 2014/11/01 · arxiv updated 2018/01/17 · crossref deposited 2022/04/08 · crossref indexed 2026/08/04
Motivated by the occurrence in rate functions of time-dependent large-deviation principles, we study a class of non-negative functions \mathscr L that induce a flow, given by \mathscr L(ρt,ρt)=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 \mathscr 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.