article published yes Filippo Quattrocchi author 3ebd6ba8-edfb-11eb-afb5-91a9745ba3080009-0000-9773-1931 JaMa department Taming Complexity in Partial Differential Systems project We prove the convergence of a modified Jordan–Kinderlehrer–Otto scheme to a solution to the Fokker–Planck equation in Ω e 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 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. Similarly to these works, we allow the mass to flow from/to the boundary ∂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 . 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 is an interval in R1, we find a formula for the descending slope of this geodesically nonconvex functional. https://research-explorer.ista.ac.at/download/20865/20945/2026_CalculusVariations_Quattrocchi.pdf application/pdfno Springer Nature2026 eng 0944-2669 1432-0835 2403.0780310.1007/s00526-025-03193-1 651 https://research-explorer.ista.ac.at/record/20571 Quattrocchi F. 2026. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. Calculus of Variations and Partial Differential Equations. 65(1), 23. Quattrocchi F. Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>Calculus of Variations and Partial Differential Equations</i>. 2026;65(1). doi:<a href="https://doi.org/10.1007/s00526-025-03193-1">10.1007/s00526-025-03193-1</a> Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature, 2026. <a href="https://doi.org/10.1007/s00526-025-03193-1">https://doi.org/10.1007/s00526-025-03193-1</a>. F. Quattrocchi, “Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions,” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 1. Springer Nature, 2026. Quattrocchi, F. (2026). Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions. <i>Calculus of Variations and Partial Differential Equations</i>. Springer Nature. <a href="https://doi.org/10.1007/s00526-025-03193-1">https://doi.org/10.1007/s00526-025-03193-1</a> Quattrocchi, Filippo. “Variational Structures for the Fokker-Planck Equation with General Dirichlet Boundary Conditions.” <i>Calculus of Variations and Partial Differential Equations</i>, vol. 65, no. 1, 23, Springer Nature, 2026, doi:<a href="https://doi.org/10.1007/s00526-025-03193-1">10.1007/s00526-025-03193-1</a>. F. Quattrocchi, Calculus of Variations and Partial Differential Equations 65 (2026). 208652025-12-29T12:06:26Z2026-04-07T08:37:46Z