Condensed Matter Physics, 2008, vol. 11, No. 4(56), p. 701-721

Title: Non-equilibrium stochastic dynamics in continuum: The free case
  Y.Kondratiev (Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany; Institute of Mathematics, Kiev, Ukraine; BiBoS, Univ. Bielefeld, Germany) ,
  E.Lytvynov (University of Wales Swansea, Singleton Park, Swansea SA2 8PP, U.K.) ,
  M.Röckner (Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D-33501 Bielefeld, Germany; BiBoS, Univ. Bielefeld, Germany)

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 α≥1, we consider the set Θα of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from Θα, will never leave Θα, 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=Rd, then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics.

Key words: birth and death process, Brownian motion on the configuration space, continuous system, Glauber dynamics, independent infinite particle process, Kawasaki dynamics, Poisson measure
PACS: 02.50.Ey, 02.50.Ga

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