Calculus of Variations and Geometric Measure Theory

V. Ferrari - F. Santambrogio

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

created by santambro on 15 Jul 2020
modified on 28 Sep 2020


Accepted Paper

Inserted: 15 jul 2020
Last Updated: 28 sep 2020

Journal: Appl. Math. Lett.
Year: 2020


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 $\rho_k\in\mathcal{P}(\Omega)$. Supposing that $V$ is $\alpha$-convex, i.e. $D^2V\geq \alpha I$, we prove that the Lipschitz constant of $\log\rho+V$ satisfies the following inequality: $\mathrm{Lip}(\log(\rho_{k+1})+V)(1+\alpha \tau)\leq \mathrm{Lip}(\log(\rho_{k})+V)$. 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.