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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ω</mml:mi> <mml:mo>⋐</mml:mo> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mi>d</mml:mi></mml:msup> </mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> is an interval in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mn>1</mml:mn></mml:msup> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>∂</mml:mi> <mml:mi>Ω</mml:mi></mml:mrow> </mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mover><mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo></mml:mover> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>Ω</mml:mi></mml:math> is an interval in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow> <mml:mn>1</mml:mn></mml:msup> </mml:math> , we find a formula for the descending slope of this geodesically nonconvex functional.