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

TL;DR

This paper establishes Lipschitz estimates for the JKO scheme applied to the Fokker-Planck equation with a semi-convex potential on bounded convex domains, demonstrating exponential decay and boundedness over time.

Contribution

It extends Lipschitz regularity results for the JKO scheme to bounded convex domains with semi-convex potentials, generalizing previous periodic case analyses.

Findings

Lipschitz constant of log ρ + V decreases exponentially when α > 0.

Lipschitz bounds are maintained over bounded time intervals.

Results align with continuous-time Fokker-Planck equation behavior.

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.