PUB - Publikationen an der Universität Bielefeld

We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t > 0. We assume that X has infinite volume and, for each alpha >= 1, we consider the set Theta(alpha) of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the alpha-th power of the volume of the set. We find quite general conditions on the process All which guarantee that the corresponding infinite particle process can start at each configuration from circle minus(alpha) will never leave circle minus(alpha) and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X = R-d, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.