Brenier-Schr{\"o}dinger problem on compact manifold with boundary

TL;DR

This paper investigates the Brenier-Schr{"o}dinger problem on compact manifolds with boundary, establishing links to Navier-Stokes equations and expanding the class of models with unique solutions using reflection group techniques.

Contribution

It extends the Brenier-Schr{"o}dinger problem framework to manifolds with boundary and introduces a method involving reflection groups to ensure uniqueness of solutions.

Findings

Link between regular solutions and Navier-Stokes equations with impermeability.

Enhanced model class with guaranteed unique solutions.

Use of quotient methods by reflection groups with multiple examples.

Abstract

We consider the Brenier-Schr{\"o}dinger problem on compact manifolds with boundary. In the spirit of a work by Arnaudon, Cruzeiro, L{\'e}onard and Zambrini, we study the kinetic property of regular solutions and obtain a link to the Navier-Stokes equations with an impermeability condition. We also enhance the class of models for which the problem admits a unique solution. This involves a method of taking quotients by reflection groups for which we give several examples.