A weak convergence approach to the large deviations of the dynamic Schr\"odinger problem

TL;DR

This paper develops a weak convergence approach to establish large deviation principles for dynamic Schr"odinger problems, extending previous results beyond Brownian motion references and introducing a uniform Laplace principle for bridge processes.

Contribution

It introduces a variational approach to large deviations for dynamic Schr"odinger problems with general reference dynamics, beyond Brownian motion, and develops a uniform Laplace principle for bridge processes.

Findings

Large deviation principles for dynamic Schr"odinger problems derived.

A uniform Laplace principle for bridge processes established.

Framework suggests potential extensions to more complex reference dynamics.

Abstract

In this paper, we consider the large deviations for dynamical Schr\"odinger problems, using the variational approach developed by Dupuis, Ellis, Budhiraja, and others. Recent results on scaled families of Schr\"odinger problems, in particular by Bernton, Ghosal, and Nutz, and the authors, have established large deviation principles for the static problem. For the dynamic problem, only the case with a scaled Brownian motion reference process has been explored by Kato. Here, we derive large deviations results using the variational approach, with the aim of going beyond the Brownian reference dynamics considered by Kato. Specifically, we develop a uniform Laplace principle for bridge processes conditioned on their endpoints. When combined with existing results for the static problem, this leads to a large deviation principle for the corresponding (dynamic) Schr\"odinger bridge. In addition to the specific results of the paper, our work puts such large deviation questions into the weak convergence framework, and we conjecture that the results can be extended to cover also more involved types of reference dynamics. Specifically, we provide an outlook on applying the result to reflected Schr\"odinger bridges.