Collective Dynamics in Biological and Social Systems

Convergence of Stochastic Interacting Particle Systems in Probability under a Sobolev Norm

Yuan Zhang



In this research, we consider the interacting particle systems with aggregation and Brownian motion. We prove that when the initial vertices are on lattice points $hiin R^d$ with mass $ ho_0(hi) h^d$, where $ ho_0$ is some initial density function, then the regularized empirical measure of the interacting particle system converges in probability to the corresponding mean-field partial differential equation with initial density $ ho_0$, under the Sobolev norm of $L^infty(L^2)cap L^2(H^1)$. Our result is true for all those systems when the aggregation, i.e., the interacting function is bounded and uniformly Lipschitz continuous over space. And for the regularized interacting particle system, it also holds for some of the most important systems whose the the interacting functions are not. For systems with Coulomb interaction, this convergence holds globally on any interval $[0,t]$. And for systems with Newton potential as interacting function, we have convergence within the largest existence time of the regular solution of the corresponding Keller-Segel equation.