On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
- Mielke, Alexander
- Peletier, Mark A.
- Renger, D. R. Michiel
2010 Mathematics Subject Classification
- 35Q82 35Q84 49S05 60F10 60J25 60J27
- Generalized gradient flows, large deviations, convex analysis, particle systems
Motivated by the occurence 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(zt,żt)=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.
- Potential Anal., 41 (2014) pp. 1293--1325.