TY - JOUR

T1 - The mean field Schrödinger problem

T2 - Ergodic behavior, entropy estimates and functional inequalities

AU - Backhoff-Veraguas, Julio

AU - Conforti, Giovanni

AU - Gentil, Ivan

AU - Léonard, Christian

N1 - Springer deal

PY - 2020/10

Y1 - 2020/10

N2 - We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian 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.

AB - We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian 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.

KW - UT-Hybrid-D

U2 - 10.1007/s00440-020-00977-8

DO - 10.1007/s00440-020-00977-8

M3 - Article

SN - 0178-8051

VL - 178

SP - 475

EP - 530

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

ER -