Strong $L^2 H^2$ convergence of the JKO scheme for the Fokker-Planck equation

TL;DR

This paper proves strong convergence in $L^2_tH^2_x$ for the JKO scheme approximating the Fokker-Planck equation, under certain regularity and domain assumptions, improving previous convergence results.

Contribution

It establishes the first strong $L^2_tH^2_x$ convergence result for the JKO scheme solving the Fokker-Planck equation, under specific regularity conditions.

Findings

Convergence is strong in $L^2_tH^2_x$ for the JKO scheme.

Assumptions include bounded, smooth convex domain and regular initial data.

The technique uses inequalities derived from optimal transport methods.

Abstract

Following a celebrated paper by Jordan, Kinderleherer and Otto it is possible to discretize in time the Fokker-Planck equation $\partial_t\varrho=\Delta\varrho+\nabla\cdot(\rho\nabla V)$ by solving a sequence of iterated variational problems in the Wasserstein space, and the sequence of piecewise constant curves obtained from the scheme is known to converge to the solution of the continuous PDE. This convergence is uniform in time valued in the Wasserstein space and also strong in $L^1$ in space-time. We prove in this paper, under some assumptions on the domain (a bounded and smooth convex domain) and on the initial datum (which is supposed to be bounded away from zero and infinity and belong to $W^{1,p}$ for an exponent $p$ larger than the dimension), that the convergence is actually strong in $L^2_tH^2_x$, hence strongly improving the previously known results in terms of the order of derivation in space. The technique is based on some inequalities, obtained with optimal transport techniques, that can be proven on the discrete sequence of approximate solutions, and that mimic the corresponding continuous computations.