Evolution of States of a Continuum Jump Model with Attraction

TL;DR

This paper investigates a continuum jump model with attraction, analyzing the evolution of states via correlation functions and proving the existence of unique solutions under certain conditions, addressing the compatibility of sub-Poissonicity with attraction.

Contribution

The paper introduces a hierarchical approach to analyze the evolution of states in an attractive jump process and proves the existence of unique classical solutions under natural conditions.

Findings

Existence of a unique classical sub-Poissonian solution on a bounded time interval.

Partial insight into the compatibility of sub-Poissonicity with attraction.

Discussion of potential approaches for a complete understanding.

Abstract

We study a model of an infinite system of point particles in $\mathds{R}^d$ performing random jumps with attraction. The system's states are probability measures on the space of particle configurations, and their evolution is described by means of Kolmogorov and Fokker-Planck equations. Instead of solving these equations directly we deal with correlation functions evolving according to a hierarchical chain of differential equations, derived from the Kolmogorov equation. Under quite natural conditions imposed on the jump kernels -- and analyzed in the paper -- we prove that this chain has a unique classical sub-Poissonian solution on a bounded time interval. This gives a partial answer to the question whether the sub-Poissonicity is consistent with any kind of attraction. We also discuss possibilities to get a complete answer to this question.