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

TL;DR

This paper establishes new limit theorems for Brownian motion that extend classical large deviation principles, with applications to backward stochastic differential equations, Schr"odinger problems, and mean field control, offering novel theoretical insights and computational schemes.

Contribution

It introduces non-exponential large deviation analogues for Wiener space and applies them to unify control and probabilistic approaches in Schr"odinger problems and stochastic PDEs.

Findings

Derived non-exponential limit theorems for Brownian motion.

Established new scaling limits for BSDEs and PDEs.

Unified control-theoretic and probabilistic approaches to Schr"odinger problems.

Abstract

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.