发表机构
Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究含空间快速振荡的 McKean-Vlasov 方程的 Wasserstein 梯度流，证明其均匀化极限为有效介质中的梯度流，并给出对不变测度的指数收敛速率，推广了线性 Fokker-Planck 方程的结果。
AI 中文摘要
我们考虑一个 McKean-Vlasov 方程，该方程在模型的能量部分和运动学部分都引入了空间快速振荡行为。前者由约束势给出，后者由底层度量给出。该方程被表述为在赋予 Wasserstein 度量的概率测度空间中的梯度流。我们确定了极限动力学，该动力学也被描述为有效介质中的梯度流。我们进一步证明了解对不变测度的长时间指数收敛。收敛速率用对数 Sobolev 常数表示，该常数关于振荡长度尺度是一致的。我们的方法基于变分不等式和泛函不等式。本文的均匀化结果将作者先前的工作 [GaoYip] 关于线性 Fokker-Planck 方程的结果推广到了非线性和非局部情形。
英文摘要
We consider a McKean-Vlasov equation incorporating spatial fast oscillatory behavior into both the energetic and kinematic components of the model. The former is given in terms of a confining potential while the latter by an underlying metric. The equation is formulated as a gradient flow in the space of probability measures endowed with a Wasserstein metric. We identify the limiting dynamics which is also described as a gradient flow in an effective media. We further prove the long time exponential convergence of solutions to the invariant measure. The rate of convergence is expressed in terms of a Logarithmic Sobolev constant which is uniform in terms of the oscillatory length scale. Our approach is based on variational and functional inequalities. The homogenization result of the current paper extends the authors' previous work [GaoYip] on a linear Fokker-Planck equation to a nonlinear and nonlocal setting.
Comments41 pages
