Lipschitz estimates on the JKO scheme for the Fokker–Planck equation on bounded convex domains

Abstract

Given a semi-convex potential V on a convex and bounded domain Ω, we consider the Jordan–Kinderlehrer–Otto scheme for the Fokker–Planck equation with potential V, which defines, for fixed time step τ>0, a sequence of densities ρk∈P(Ω). Supposing that V is α-convex, i.e. D2V≥αI, we prove that the Lipschitz constant of logρ+V satisfies the following inequality: Lip(log(ρk+1)+V)(1+ατ)≤Lip(log(ρk)+V). This provides exponential decay if α>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.