论文标题
fokker-planck方程的进化$γ$ - 多个维度的熵梯度流量结构
Evolutionary $Γ$-convergence of entropic gradient flow structures for Fokker-Planck equations in multiple dimensions
论文作者
论文摘要
我们考虑$ \ Mathbb {r}^d $中有限凸域上Fokker-Planck方程的有限体积近似值，并研究相应的梯度流量结构。我们通过进化$γ$ - convergence的方法来谴责离散到连续的fokker-planck方程，即我们在梯度流量结构的水平上传递到极限，从而概括了由异见和liero获得的一维结果。证明具有变异性质，并依赖于具有独立感兴趣的离散到核电限制的功能的MOSCO收敛结果。我们的结果适用于任意的常规网格，即使相关的离散运输距离可能无法在此通用性中汇聚到Wasserstein距离。
We consider finite-volume approximations of Fokker-Planck equations on bounded convex domains in $\mathbb{R}^d$ and study the corresponding gradient flow structures. We reprove the convergence of the discrete to continuous Fokker-Planck equation via the method of Evolutionary $Γ$-convergence, i.e., we pass to the limit at the level of the gradient flow structures, generalising the one-dimensional result obtained by Disser and Liero. The proof is of variational nature and relies on a Mosco convergence result for functionals in the discrete-to-continuum limit that is of independent interest. Our results apply to arbitrary regular meshes, even though the associated discrete transport distances may fail to converge to the Wasserstein distance in this generality.
