Large deviations for dynamical Schrödinger problems

We establish large deviations for dynamical Schrödinger problems driven by perturbed Brownian motions when the noise parameter tends to zero. Our results show that Schrödinger bridges charge exponentially small masses outside of the support of the limiting law that agrees with the optimal solution to the dynamical Monge-Kantorovich optimal transport problem. Our proofs build on mixture representations of Schrödinger bridges and establishing exponential continuity of Brownian bridges with respect to the initial and terminal points.