Evolutionary $Γ$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions

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.

Homogenisation of dynamical optimal transport on periodic graphs

“…The main result in [GKM20] asserts that, for a sequence of meshes on a bounded convex domain in R d , an isotropy condition on the meshes is required to obtain the convergence of the discrete dynamical transport distances to W 2 . This is in sharp contrast to the scaling behaviour of the corresponding gradient flow dynamics, for which no additional symmetry on the meshes is required to ensure the convergence of discretised evolution equations to the expected continuous limit [DiL15,FMP20].…”

Diffusive limit of random walks on tessellations via generalized gradient flows

“…Once we have identified the limit energy-dissipation functional , we can make use of classical gradient flow theory to deduce the form of the limit forward Kolmogorov equation (fK). In particular, we formally obtain the diffusion equation The techniques we use to prove the lim inf inequalities in Step 2 are similar to those used in [21]; however, the philosophy and results have considerable differences. The authors of [21] prove the convergence of the finite-volume discretization of the equation (1.1) with = I to the original equation.…”

“…In particular, we formally obtain the diffusion equation The techniques we use to prove the lim inf inequalities in Step 2 are similar to those used in [21]; however, the philosophy and results have considerable differences. The authors of [21] prove the convergence of the finite-volume discretization of the equation (1.1) with = I to the original equation. We, on the other hand, start with a more general discrete evolution equation (fK ℎ ) and, consequently, recover the diffusion equation (1.1) with variable coefficients .…”

“…These methods are known to converge for a restrictive class of tessellations. From the variational perspective, an approach similar to ours was used in [15] (one-dimension) and [21] (multi-dimension) to prove convergence of the finite-volume method for the Fokker-Planck equation.…”

“…Another well-studied choice is the quadratic gradient structure. In particular, the quadratic structure was used to prove the convergence for the finite-volume discretization of the Fokker-Planck equation in [21]. In comparison, the adoption of the cosh-type gradient structure allows us to consider a more general class of tessellations, including a tilted ℤ tessellation (Example 2.9), and we can dispense with the orthogonality assumption used in [21] and the finite-volume methods mentioned above.…”