Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions.

Abstract

We prove the convergence of a modified Jordan-Kinderlehrer-Otto scheme to a solution to the Fokker-Planck equation in  $\Omega ⋐R^d$ with general-strictly positive and temporally constant-Dirichlet boundary conditions. We work under mild assumptions on the domain, the drift, and the initial datum. In the special case where  $\Omega$ is an interval in  $R^1$ , we prove that such a solution is a gradient flow-curve of maximal slope-within a suitable space of measures, endowed with a modified Wasserstein distance. Our discrete scheme and modified distance draw inspiration from contributions by A. Figalli and N. Gigli [J. Math. Pures Appl. 94, (2010), pp. 107-130], and J. Morales [J. Math. Pures Appl. 112, (2018), pp. 41-88] on an optimal-transport approach to evolution equations with Dirichlet boundary conditions. Similarly to these works, we allow the mass to flow from/to the boundary  $\partial \Omega$ throughout the evolution. However, our leading idea is to also keep track of the mass at the boundary by working with measures defined on the whole closure  $\overline{\Omega }$ . The driving functional is a modification of the classical relative entropy that also makes use of the information at the boundary. As an intermediate result, when  $\Omega$ is an interval in  $R^1$ , we find a formula for the descending slope of this geodesically nonconvex functional.