论文标题
具有单数内核的非线性fokker-Planck方程的定量粒子近似
Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel
论文作者
论文摘要
在这项工作中，我们研究了中等相互作用的粒子系统与奇异相互作用内核的经验度量的收敛性。首先，我们证明了粒子位置的经验度量的时间边缘朝向限制非线性fokker-Planck方程的定量收敛。其次，我们证明了McKean-Vlasov SDE的适当性，涉及这种奇异内核以及经验量度对其的融合（混乱的传播）。 我们的结果只需要在相互作用内核上（包括生物 - 萨瓦特内核）以及有吸引力的内核（例如Riesz和Keller-segel内核）中的相互作用内核的规律性非常弱。对于其中一些重要示例，这是第一次通过随机粒子系统获得PDE的定量近似。特别是，这种收敛仍然（及时在本地）对于在有限时间内表现出爆炸的PDE。 这些证明是基于半群方法的方法，结合了对无限多维随机卷积积分的规律性的精细分析。
In this work, we study the convergence of the empirical measure of moderately interacting particle systems with singular interaction kernels. First, we prove quantitative convergence of the time marginals of the empirical measure of particle positions towards the solution of the limiting nonlinear Fokker-Planck equation. Second, we prove the well-posedness for the McKean-Vlasov SDE involving such singular kernels and the convergence of the empirical measure towards it (propagation of chaos). Our results only require very weak regularity on the interaction kernel, including the Biot-Savart kernel, and attractive kernels such as Riesz and Keller-Segel kernels in arbitrary dimension. For some of these important examples, this is the first time that a quantitative approximation of the PDE is obtained by means of a stochastic particle system. In particular, this convergence still holds (locally in time) for PDEs exhibiting a blow-up in finite time. The proofs are based on a semigroup approach combined with a fine analysis of the regularity of infinite-dimensional stochastic convolution integrals.
