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) $\Vert k_{{\mu }_0}^{(n)}{\Vert }_{L^{\infty }(({\mathbb{R}}^d{)}^n)}\le C^n$, $n\in \mathbb{N}$ (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 ${\mu }_t$.

Proceedings of the Conference Complex Analysis and Dynamical Systems VI, Naharyia, 2013