On the relation between gradient flows and the largedeviation principle, with applications to Markov chains and diffusion
Abstract
Motivated by the occurrence in rate functions of timedependent largedeviation principles, we study a class of nonnegative functions $\mathscr L$ that induce a flow, given by $\mathscr L(\rho_t,\dot\rho_t)=0$. 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 $\mathscr L$ 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 entropyWasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.
 Publication:

arXiv eprints
 Pub Date:
 December 2013
 arXiv:
 arXiv:1312.7591
 Bibcode:
 2013arXiv1312.7591M
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 Mathematics  Analysis of PDEs;
 Mathematics  Probability;
 35Q82;
 35Q84;
 49S05;
 60F10;
 60J25;
 60J27
 EPrint:
 Potential Analysis 41:4 (2014) 12931327