preprint

Inserted: 13 mar 2024

Year: 2024

Abstract:

We prove convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in $\Omega \Subset \mathbb{R}^d$ with spatially nonconstant Dirichlet boundary conditions. We work under mild assumptions on the domain, on the drift, and on 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.