@article{11700,
  abstract     = {This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou–Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport. Secondly, in the spirit of Jordan–Kinderlehrer–Otto, we show that McKean–Vlasov equations can be formulated as gradient flow of the free energy in the Wasserstein space of probability measures. The proofs of these results are based on careful regularisation arguments to circumvent some of the difficulties arising in metric graphs, namely, branching of geodesics and the failure of semi-convexity of entropy functionals in the Wasserstein space.},
  author       = {Erbar, Matthias and Forkert, Dominik L and Maas, Jan and Mugnolo, Delio},
  issn         = {1556-181X},
  journal      = {Networks and Heterogeneous Media},
  number       = {5},
  pages        = {687--717},
  publisher    = {American Institute of Mathematical Sciences},
  title        = {{Gradient flow formulation of diffusion equations in the Wasserstein space over a metric graph}},
  doi          = {10.3934/nhm.2022023},
  volume       = {17},
  year         = {2022},
}

@article{11739,
  abstract     = {We consider finite-volume approximations of Fokker--Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker--Planck equation via the method of evolutionary $\Gamma$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalizing the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.},
  author       = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo},
  issn         = {1095-7154},
  journal      = {SIAM Journal on Mathematical Analysis},
  keywords     = {Fokker--Planck equation, gradient flow, evolutionary $\Gamma$-convergence},
  number       = {4},
  pages        = {4297--4333},
  publisher    = {Society for Industrial and Applied Mathematics},
  title        = {{Evolutionary $\Gamma$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}},
  doi          = {10.1137/21M1410968},
  volume       = {54},
  year         = {2022},
}

@phdthesis{7629,
  abstract     = {This thesis is based on three main topics: In the first part, we study convergence of discrete gradient flow structures associated with regular finite-volume discretisations of Fokker-Planck equations. We show evolutionary I convergence of the discrete gradient flows to the L2-Wasserstein gradient flow corresponding to the solution of a Fokker-Planck
equation in arbitrary dimension d >= 1. Along the argument, we prove Mosco- and I-convergence results for discrete energy functionals, which are of independent interest for convergence of equivalent gradient flow structures in Hilbert spaces.
The second part investigates L2-Wasserstein flows on metric graph. The starting point is a Benamou-Brenier formula for the L2-Wasserstein distance, which is proved via a regularisation scheme for solutions of the continuity equation, adapted to the peculiar geometric structure of metric graphs. Based on those results, we show that the L2-Wasserstein space over a metric graph admits a gradient flow which may be identified as a solution of a Fokker-Planck equation.
In the third part, we focus again on the discrete gradient flows, already encountered in the first part. We propose a variational structure which extends the gradient flow structure to Markov chains violating the detailed-balance conditions. Using this structure, we characterise contraction estimates for the discrete heat flow in terms of convexity of
corresponding path-dependent energy functionals. In addition, we use this approach to derive several functional inequalities for said functionals.},
  author       = {Forkert, Dominik L},
  issn         = {2663-337X},
  pages        = {154},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Gradient flows in spaces of probability measures for finite-volume schemes, metric graphs and non-reversible Markov chains}},
  doi          = {10.15479/AT:ISTA:7629},
  year         = {2020},
}

@unpublished{10022,
  abstract     = {We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in R^d and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary Γ-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.},
  author       = {Forkert, Dominik L and Maas, Jan and Portinale, Lorenzo},
  booktitle    = {arXiv},
  pages        = {33},
  title        = {{Evolutionary Γ-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions}},
  year         = {2020},
}