Free Jump Dynamics in Continuum

TL;DR

This paper studies the evolution of an infinite system of hopping particles in continuous space, establishing conditions under which the correlation functions evolve smoothly and correspond to unique system states over time.

Contribution

It introduces a framework for the continuous differentiable evolution of correlation functions for infinite particle systems with sub-Poissonian initial states.

Findings

Correlation functions evolve as solutions to a specific evolution equation.

The evolved correlation functions correspond to unique system states.

The evolution is continuously differentiable in the space of sub-Poissonian functions.

Abstract

The evolution is described of an infinite system of hopping point particles in $\mathbb{R}^d$. The states of the system are probability measures on the space of configurations of particles. Under the condition that the initial state $\mu_0$ has correlation functions of all orders which are: (a) $k_{\mu_0}^{(n)} \in L^\infty ((\mathbb{R}^d)^n)$ (essentially bounded); (b) $\|k_{\mu_0}^{(n)}\|_{ L^\infty ((\mathbb{R}^d)^n)} \leq C^n$, $n\in \mathbb{N}$ (sub-Poissonian), the evolution $\mu_0 \mapsto \mu_t$, $t>0$, is obtained as a continuously differentiable map $k_{\mu_0} \mapsto k_t$, $k_t =(k_t^{(n)})_{n\in \mathbb{N}}$, in the space of essentially bounded sub-Poissonian functions. In particular, it is proved that $k_t$ solves the corresponding evolution equation, and that for each $t>0$ it is the correlation function of a unique state $\mu_t$.