论文标题
弗拉索夫·弗克·普朗克方程的平均场限制的新方法
A new approach to the mean-field limit of Vlasov-Fokker-Planck equations
论文作者
论文摘要
本文介绍了一种新颖的方法，以实现相互作用颗粒的随机系统的平均场限制，从而导致平均场极限对Vlasov-Poisson-Fokker-Planck系统的第一个衍生物限制在维度〜$ 2 $的等离子体中，并在尺寸〜$ 3 $中获得部分结果。该方法与导致动力学方程的二阶系统广泛兼容，并且依赖于对BBGKY层次结构的新估计。通过利用速度的扩散，这些估计值均匀地在粒子数量中均匀地结合了边缘的加权$ l^p $规范。这允许定性地得出粒子之间非常单数相互作用内核的平均场极限，包括排斥泊松相互作用，以及对$ l^2 $的一般核的定量估计。
This article introduces a novel approach to the mean-field limit of stochastic systems of interacting particles, leading to the first ever derivation of the mean-field limit to the Vlasov-Poisson-Fokker-Planck system for plasmas in dimension~$2$ together with a partial result in dimension~$3$. The method is broadly compatible with second order systems that lead to kinetic equations and it relies on novel estimates on the BBGKY hierarchy. By taking advantage of the diffusion in velocity, those estimates bound weighted $L^p$ norms of the marginals or observables of the system, uniformly in the number of particles. This allows to qualitatively derive the mean-field limit for very singular interaction kernels between the particles, including repulsive Poisson interactions, together with quantitative estimates for a general kernel in $L^2$.
