Published Paper

Inserted: 13 mar 2024
Last Updated: 6 dec 2025

Journal: Calculus of Variations and Partial Differential Equations
Volume: 65
Number: 23
Year: 2026
Doi: 10.1007/s00526-025-03193-1

Abstract:

We prove the convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\Omega \Subset \mathbb R^d$ 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 $\mathbb R^1$, 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 $\mathbb R^1$, we find a formula for the descending slope of this geodesically nonconvex functional.