*Accepted Paper*

**Inserted:** 15 jul 2020

**Last Updated:** 28 sep 2020

**Journal:** Appl. Math. Lett.

**Year:** 2020

**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 $\rho_k\in\mathcal{P}(\Omega)$. 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.

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