Optimal transportation under McKean-Vlasov dynamics

Abstract

We study a broad class of stochastic transport problems under controlled McKean-Vlasov dynamics, allowing control of both drift and volatility, with weak terminal constraints. In contrast to the classical Schrödinger problem, the reference dynamics depend on the evolving law of the process, leading to nonlinear optimization problems on path space, while the terminal constraint is formulated through weak optimal transport, permitting a flexible treatment of distributional targets. This framework encompasses several problems of current interest, including the mean field Schrödinger problem, martingale optimal transport, and Schrödinger-Bass bridges. We establish three main results. First, we prove the existence of optimal controls under general coercivity, continuity, and convexity assumptions. Second, we derive a convex dual representation that extends weak Kantorovich duality and stochastic optimal transport duality to controlled McKean-Vlasov dynamics. Finally, for the mean field Schrödinger problem, we prove convergence, at the level of value functions, of the associated finite-particle Schrödinger bridge problems to the mean field limit, thereby identifying the nonlinear problem as the exact infinite-particle limit of finite-particle Schrödinger bridges. As a byproduct, we establish the existence of an optimal mean field Schrödinger potential, providing a variational foundation for duality in nonlinear stochastic transport.