论文标题
与奇异内核的非交换相互作用扩散的平均场极限
Mean-field limit of Non-exchangeable interacting diffusions with singular kernels
论文作者
论文摘要
相互作用扩散的平均场限量，而无需交换性，由加权相互作用和非i.i.d。研究初始值。权重可以签名和无限。该结果适用于包括生物 - 萨瓦特法（Biot-Savart Law）在内的大量奇异内核。与$ \ frac {1} {n} {n} \ sum_ {i = 1}^nδ__{x_i} $相比，我们演示了一种均值的均值收敛类型。更具体地说，具有任意均匀$ l^r $ - 重量的签名经验度量过程的序列，$ r> 1 $，弱收敛到耦合的PDE，例如描述2D Navier-Stokes方程的被动标量的动力学。 我们的方法基于一个紧密的论点，并利用了系统统一的渔民信息。主要的困难是确定如何在不存在非交换情况的定义性 - 霍特 - 卫星定理的情况下传播经验度量极限的规律性。为此，通过一系列随机度量与一系列加权经验措施薄弱合并，并具有统一的Sobolev规律性，是通过粒子的关节定律的分解来构建的。
The mean-field limit of interacting diffusions without exchangeability, caused by weighted interactions and non-i.i.d. initial values, are investigated. The weights could be signed and unbounded. The result applies to a large class of singular kernels including the Biot-Savart law. We demonstrate a flexible type of mean-field convergence, in contrast to the typical convergence of $\frac{1}{N}\sum_{i=1}^Nδ_{X_i}$. More specifically, the sequence of signed empirical measure processes with arbitrary uniform $l^r$-weights, $r>1$, weakly converges to a coupled PDE's, such as the dynamics describing the passive scalar advected by the 2D Navier-Stokes equation. Our method is based on a tightness/compactness argument and makes use of the systems' uniform Fisher information. The main difficulty is to determine how to propagate the regularity properties of the limits of empirical measures in the absence of the DeFinetti-Hewitt-Savage theorem for the non-exchangeable case. To this end, a sequence of random measures, which merges weakly with a sequence of weighted empirical measures and has uniform Sobolev regularity, is constructed through the disintegration of the joint laws of particles.
