Abstract
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.
Highlights
At a given time T, the configuration of the particle system is visible to an external observer that finds it close to an “unexpected” probability measure μfin, namely
It is a classical result [4,20,49] that the sequence of empirical path measures (2) obeys the large deviations principle (LDP)
Extending naturally the classical terminology we say that an optimal path measure is a mean field Schrödinger bridge ( mean field Schrödinger bridges (MFSB)) and the optimal value is the mean field entropic cost
Summary
( , Ft , FT ) is the canonical space of Rd -valued continuous paths on [0, T ], so. In the case when W is convex, the rate function H(P| (P)) is not convex in the usual sense, the entropy F is displacement convex in the sense of McCann [38] This observation was used to prove uniqueness of minimizers for F, and could be the starting point towards uniqueness for (MFSP). Even the most recent works on large deviations for weakly interacting particle systems such as [4] do not seem to cover the setting and scope of Theorem 1.1 This is because in those references the LDPs are obtained for a topology that is weaker than the W1-topology, that is what we need later on. Problem (MFSP) has the desired interpretation of finding the most likely evolution of the particle system conditionally on the observations (when N is very large)
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