论文标题
平面中双抛物线凯勒 - 塞格方程的粒子近似
Particle approximation of the doubly parabolic Keller-Segel equation in the plane
论文作者
论文摘要
在这项工作中，我们研究了与抛物线抛物线凯勒 - 塞格系统相关的N颗粒的随机系统。该粒子系统是奇异的，而非马尔可夫人的漂移项取决于粒子的过去。当敏感性参数足够小时，我们表明该粒子系统确实存在于任何$ n \ geq 2 $时，我们在其经验度量的$ n $中表现出紧密度，并且这种经验度量的任何弱极限点，例如$ n \ to \ infty $一些微弱的感觉。证明的主要论点包括对相互作用内核的马克维亚化：我们表明，从某种意义上说，与抛物线纤维纤维纤维的情况一样，可以通过两乘二的库仑相互作用来控制两乘路径依赖的相互作用。
In this work, we study a stochastic system of N particles associated with the parabolic-parabolic Keller-Segel system. This particle system is singular and non Markovian in that its drift term depends on the past of the particles. When the sensitivity parameter is sufficiently small, we show that this particle system indeed exists for any $N\geq 2$, we show tightness in $N$ of its empirical measure, and that any weak limit point of this empirical measure, as $N\to\infty$, solves some nonlinear martingale problem, which in particular implies that its family of time-marginals solves the parabolic-parabolic Keller-Segel system in some weak sense. The main argument of the proof consists of a Markovianization of the interaction kernel: We show that, in some loose sense, the two-by-two path-dependant interaction can be controlled by a two-by-two Coulomb interaction, as in the parabolic-elliptic case.
