PUB - Publikationen an der Universität Bielefeld

The dynamics of an infinite system of point particles in R-d, which hop and interact with each other, is described at both micro- and mesoscopic levels. The states of the system are probability measures on the space of configurations of particles. For a bounded time interval , the evolution of states mu 0 bar right arrow mu(t) is shown to hold in a space of sub-Poissonian measures. This result is obtained by: (a) solving equations for correlation functions, which yields the evolution k(0) bar right arrow k(t), t is an element of {0, T) in a scale of Banach spaces; (b) proving that each is a correlation function for a unique measure mu(t). The mesoscopic theory is based on a Vlasov-type scaling, that yields a mean-field-like approximate description in terms of the particles' density which obeys a kinetic equation. The latter equation is rigorously derived from that for the correlation functions by the scaling procedure. We prove that the kinetic equation has a unique solution rho(t), t is an element of {0, +infinity).