Strong \(L^2 H^2\) Convergence of the JKO Scheme for the Fokker–Planck Equation

Following a celebrated paper by Jordan, Kinderleherer and Otto, it is possible to discretize in time the Fokker–Planck equation \(\partial _t\varrho =\Delta \varrho +\nabla \cdot (\varrho \nabla V)\) 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 open 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.