Condensed Matter Physics, 2008, vol. 11, No. 4(56), p. 701721
DOI:10.5488/CMP.11.4.701
Title:
Nonequilibrium stochastic dynamics in continuum: The free case
Author(s):

Y.Kondratiev
(Fakultät für Mathematik, Universität Bielefeld, Postfach 10 01 31, D33501 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, D33501 Bielefeld, Germany; BiBoS, Univ. Bielefeld, Germany)

We study the problem of identification of a proper statespace 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 Θ_{α} of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the αth 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 Θ_{α}, will never leave Θ_{α}, 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 birthanddeath 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 (nonequilibrium) free Glauber dynamics is a scaling limit of (nonequilibrium) 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
