Non-exponential Sanov and Schilder theorems on Wiener space: BSDEs, Schr\"odinger problems and Control

We derive new limit theorems for Brownian motion, which can be seen as non-exponential analogues of the large deviation theorems of Sanov and Schilder in their Laplace principle forms. As a first application, we obtain novel scaling limits of backward stochastic differential equations and their related partial differential equations. As a second application, we extend prior results on the small-noise limit of the Schr\"odinger problem as an optimal transport cost, unifying the control-theoretic and probabilistic approaches initiated respectively by T. Mikami and C. L\'eonard. Lastly, our results suggest a new scheme for the computation of mean field optimal control problems, distinct from the conventional particle approximation. A key ingredient in our analysis is an extension of the classical variational formula (often attributed to Borell or Bou\'e-Dupuis) for the Laplace transform of Wiener measure.