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

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