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 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 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.