Large deviations for scaled families of Schr\"odinger bridges with reflection

Abstract

In this paper, we show a large deviation principle for certain sequences of static Schr\"{o}dinger bridges, typically motivated by a scale-parameter decreasing towards zero, extending existing large deviation results to cover a wider range of reference processes. Our results provide a theoretical foundation for studying convergence of such Schr\"{o}dinger bridges to their limiting optimal transport plans. Within generative modeling, Schr\"{o}dinger bridges, or entropic optimal transport problems, constitute a prominent class of methods, in part because of their computational feasibility in high-dimensional settings. Recently, Bernton et al. established a large deviation principle, in the small-noise limit, for fixed-cost entropic optimal transport problems. In this paper, we address an open problem posed by Bernton et al. and extend their results to hold for Schr\"{o}dinger bridges associated with certain sequences of more general reference measures with enough regularity in a similar small-noise limit. These can be viewed as sequences of entropic optimal transport plans with non-fixed cost functions. Using a detailed analysis of the associated Skorokhod maps and transition densities, we show that the new large deviation results cover Schr\"{o}dinger bridges where the reference process is a reflected diffusion on bounded convex domains, corresponding to recently introduced model choices in the generative modeling literature.