论文标题
Lipschitz对有限凸形域的fokker-Plack方程的JKO方案的估计
Lipschitz estimates on the JKO scheme for the Fokker-Plack equation on bounded convex domains
论文作者
论文摘要
考虑到在凸面和有限域上具有半凸电势V，我们考虑使用带势V的fokker-Planck方程的Jordan-Kinderlehrer-otto方案，该方程定义了固定的时间步长$τ$> 0，$ρ$ k $ p（$ pec）的固定时间步骤$τ$> 0。假设V为$α$ -CONVEX，即D 2 V $ \ ge $ $ $α$ i，我们证明Lipschitz的Lipschitz常数$ρ$ + V的常数满足以下不平等：LIP（log（$ρ$ k + 1） + v）（1 + $ $ $α$α$α$ \ the $ \ le $ \ le $ lip（log $ lig（$ k） + v）。如果$α$> 0，LIPSCHITZ在有限的时间间隔内将提供指数衰减，这与连续时方程的结果相一致，并在周期性的情况下扩展了Lee先前的分析。
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 $ρ$ k $\in$ P($Ω$). Supposing that V is $α$-convex, i.e. D 2 V $\ge$ $α$I, we prove that the Lipschitz constant of log $ρ$ + V satisfies the following inequality: Lip(log($ρ$ k+1) + V)(1 + $α$$τ$) $\le$ Lip(log($ρ$ k) + 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.
