Brenier-Schrodinger problem on compact manifolds with boundary

We consider the Brenier-Schrodinger problem on compact manifolds with boundary. In the spirit of a work by Arnaudon, Cruzeiro, Leonard, and Zambrini, we study the kinetic property of regular solutions and obtain a link to the Navier-Stokes equations with an impermeability condition. Moreover, by a method of taking quotients by reflection groups, we enhance the class of models for which the problem admits a unique solution.