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

We prove convergence of a modified Jordan--Kinderlehrer--Otto scheme to a solution to the Fokker--Planck equation in Ω Rᵈ 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 Ω is an interval in R¹, 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.