The mean field Schrodinger problem: ergodic behavior, entropy estimates and functional inequalities

• 成果类型: Article; Early Access
• 署名作者: Backhoff, Julio; Conforti, Giovani; Gentil, Ivan; Leonard, Christian
• 署名单位: University of Vienna; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); Ecole Centrale de Lyon; Institut National des Sciences Appliquees de Lyon - INSA Lyon; Universite Claude Bernard Lyon 1; Universite Jean Monnet; University of Twente
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00977-8
• 发表日期: 2020
• 关键词: long-time average; Optimal Transport; granular media; EQUATIONS; games; cost

摘要

We study the mean field Schrodinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud ofinteractingBrownian particles conditionally on the observation of their initial and final configuration. Its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles. We undertake a fine study of the dynamics of its solutions, including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the time between observations grows larger and larger, as well as a novel class of functional inequalities involving the mean field entropic cost (i.e. the optimal value in (MFSP)). Our strategy unveils an interesting connection between forward backward stochastic differential equations and the Riemannian calculus on the space of probability measures introduced by Otto, which is of independent interest.