TY - JOUR

T1 - On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion

AU - Mielke, A.

AU - Renger, D.R.M.

AU - Peletier, M.A.

PY - 2014

Y1 - 2014

N2 - 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(¿t,¿¿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 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.
Keywords: Generalized gradient flows · Large deviations · Convex analysis · Particle systems

AB - 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(¿t,¿¿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 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.
Keywords: Generalized gradient flows · Large deviations · Convex analysis · Particle systems

U2 - 10.1007/s11118-014-9418-5

DO - 10.1007/s11118-014-9418-5

M3 - Article

VL - 41

SP - 1293

EP - 1327

JO - Potential Analysis

JF - Potential Analysis

SN - 0926-2601

IS - 4

ER -