An infinite system of point particles performing random jumps in R d with repulsion is studied. The states of the system are probability measures on the space of particle's configurations. The result of the paper is the construction of the global in time evolution of states with the help of the corresponding correlation functions. It is proved that for each initial sub-Poissonian state µ0, the constructed evolution µ0 → µt preserves this property. That is, µt is sub-Poissonian for all t > 0.1991 Mathematics Subject Classification. 82C22; 70F45; 60K35.

We study the dynamics of an infinite system of point particles of two types. They perform random jumps in R d in the course of which particles of different types repel each other whereas those of the same type do not interact. The states of the system are probability measures on the corresponding configuration space. The main result is the construction of the global (in time) Markov evolution of such states by means of correlation functions. It is proved that for each initial sub-Poissonian state μ 0 , the states evolve μ 0 → μ t in such a way that μ t is sub-Poissonian for all t > 0. The mesoscopic (approximate) description of the evolution of states is also given. The stability of translation invariant stationary states is studied. In particular, we show that some of such states can be unstable with respect to space-dependent perturbations.

Abstract. The evolution is described of an infinite system of hopping point particles in R d . The states of the system are probability measures on the space of configurations of particles. Under the condition that the initial state µ0 has correlation functions of all orders which are:, the evolution µ0 → µt, t > 0, is obtained as a continuously differentiable map kµ 0 → kt, kt = (k (n) t ) n∈N , in the space of essentially bounded sub-Poissonian functions. In particular, it is proved that kt solves the corresponding evolution equation, and that for each t > 0 it is the correlation function of a unique state µt.

No abstract