arXiv 2011.005372020-11math.PRmath.AP

用singular kernel对非线性Fokker-Planck方程的定量粒子近似

Quantitative particle approximation of nonlinear Fokker-Planck equations with singular kernel

Christian Olivera, Alexandre Richard, Milica Tomasevic

摘要：

在这项工作中，我们研究了具有奇异相互作用核的适度相互作用粒子系统的经验测度的收敛性。首先，我们证明了定量收敛的时间边缘的粒子位置的经验措施的解决方案的限制非线性福克-普朗克方程。其次，我们证明了涉及这种奇异核的McKean-Vlasov矩阵的适定性和经验测度的收敛性（混沌的传播）。我们的结果只需要非常弱的规律性的相互作用内核，包括Biot-Savart内核，和有吸引力的内核，如Riesz和Keller-Segel内核在任意维度。对于这些重要的例子中的一些，这是第一次，通过随机粒子系统获得的PDE的定量近似。特别是，这种收敛仍然保持（局部时间）表现出爆破在有限时间的偏微分方程。证明是基于一个半群的方法结合了精细的分析的规则性的无穷维随机卷积积分。

英文摘要：

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.

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摘要：

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