Strong \(L^2 H^2\) convergence of the JKO scheme for the Fokker-Planck equation (Q6634213)

Consider the following linear Fokker-Planck equation on a bounded smooth convex domain \N\[\N\partial_t \rho_t= \Delta \rho_t + \nabla\cdot(\rho_t\nabla V)\N\]\Nfor a nice potential \(V\), and consider the time discrete scheme of this equation by solving a sequence of iterated variational problems in the Wasserstein space. It was shown in this paper that under some assumptions on the initial datum, the time discrete scheme converges strongly in \(L^2_t H_x^2,\) which improves the existing results in terms of the order of derivation in space.