@article{20865,
  abstract     = {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.},
  author       = {Quattrocchi, Filippo},
  issn         = {1432-0835},
  journal      = {Calculus of Variations and Partial Differential Equations},
  number       = {1},
  publisher    = {Springer Nature},
  title        = {{Variational structures for the Fokker-Planck equation with general Dirichlet boundary conditions}},
  doi          = {10.1007/s00526-025-03193-1},
  volume       = {65},
  year         = {2026},
}