Free Jump Dynamics in Continuum

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 $\mu_t$.