A PDE approach to Benamou--Brenier formula for the Schr\"odinger problem

TL;DR

This paper extends the Benamou--Brenier formulation of the Schr"odinger problem to sub-Gaussian measures, enabling its application to more general, unbounded support scenarios like Gaussian models.

Contribution

It broadens the validity of the Benamou--Brenier formula to sub-Gaussian measures, relaxing the compact support assumption in existing proofs.

Findings

Established existence of velocity fields with polynomial growth.

Provided an almost self-contained proof using Hessian estimates.

Justified dynamic formulation use in Gaussian and mixture models.

Abstract

We studied the Benamou--Brenier formulation of the Schr\"odinger problem, focusing on a gap between theoretical results and applications, that often involve measures with unbounded support. While the existing proof in the literature relies on the compactness of the marginals' supports to ensure the necessary regularity of the Schr\"odinger potentials, we extend the validity of the Benamou--Brenier formula to the larger class of sub-Gaussian probability measures. Exploiting fine estimates on the Hessian of the potentials and the entropic interpolation, we provide an almost self-contained proof that establishes the existence of a velocity field with the appropriate polynomial growth that ensures the right integrability. This result justifies the use of the dynamic formulation in more general settings, such as Gaussian and mixture-of-Gaussians models, important also for the applications.