Evolution of states of an infinite particle system with nonlocal branching

TL;DR

This paper analyzes the evolution of an infinite particle system with nonlocal branching by solving associated nonlinear and linear equations, establishing existence and uniqueness of solutions for the system's state evolution.

Contribution

It introduces a novel approach using tempered counting measures and solves the Fokker-Planck equation for infinite systems with nonlocal branching.

Findings

Unique solution to the Fokker-Planck equation for the system

Method for passing to tempered counting measures

Solution of nonlinear log-Laplace and Kolmogorov equations

Abstract

We study the evolution of states of an infinite system of point particles dwelling in a locally compact Polish space $X$. Each particle produces at random a finite `cloud' of offsprings distributed over $X$ according to some law, and disappears afterwards. The system's states are probability measures on an appropriate space of locally finite counting measures on $X$. Their evolution is obtained by solving the corresponding Fokker-Planck equation. We prove that this equation has a unique solution and discuss some of its properties. Our pivotal idea of dealing with infinite systems consists in passing to tempered counting measures by imposing appropriate restrictions on the branching. In this approach, we first solve a nonlinear evolution equation in the space of bounded continuous functions on $X$ -- so called log-Laplace equation. Next we solve the Kolmogorov equation which is then used to solve the Fokker-Planck equation and thus describe the evolution in question.