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 $\mathscr L$ 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 entropy-Wasserstein gradient structure, and for independent Markovian particles on a finite state space, which leads to a previously unknown gradient structure.
- Publication:
-
arXiv e-prints
- Pub Date:
- December 2013
- DOI:
- 10.48550/arXiv.1312.7591
- 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
- E-Print:
- Potential Analysis 41:4 (2014) 1293-1327