Lipschitz estimates on the JKO scheme for the Fokker-Plack equation on bounded convex domains

Abstract

Given a semi-convex potential V on a convex and bounded domain $\Omega$, we consider the Jordan-Kinderlehrer-Otto scheme for the Fokker-Planck equation with potential V, which defines, for fixed time step $\tau$ > 0, a sequence of densities $\rho$ k $\in$ P($\Omega$). Supposing that V is $\alpha$-convex, i.e. D 2 V $\ge$ $\alpha$I, we prove that the Lipschitz constant of log $\rho$ + V satisfies the following inequality: Lip(log($\rho$ k+1) + V)(1 + $\alpha$$\tau$) $\le$ Lip(log($\rho$ k) + V). This provides exponential decay if $\alpha$ > 0, Lipschitz bounds on bounded intervals of time, which is coherent with the results on the continuous-time equation, and extends a previous analysis by Lee in the periodic case.