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.

中文翻译：

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平均场薛定谔问题：遍历行为、熵估计和函数不等式

我们研究了平均场薛定谔问题 (MFSP)，即根据观察到的布朗粒子的初始和最终配置，有条件地找到最可能的相互作用的布朗粒子云演化的问题。其严格的公式是关于具有边际约束的优化问题，其目标函数是与弱相关布朗粒子系统相关的大偏差率函数。我们对其解的动力学进行了精细的研究，包括定量能量耗散估计，随着观测之间的时间越来越长，产生指数收敛到平衡，以及一类新的涉及平均场熵成本的函数不等式（即(MFSP) 中的最佳值）。