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

We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in Ω Rᵈ Ω ⋐ 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 Ω Ω is an interval in R¹ 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 ∂ Ω ∂ Ω 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 Ω ̄ Ω ¯ . 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 Ω Ω is an interval in R¹ R 1 , we find a formula for the descending slope of this geodesically nonconvex functional.