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 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 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 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 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. 2000

A lattice model of interacting light quantum particles of mass m oscillating in a crystalline field is considered in the framework of an approach based on functional integrals. The main aspects of this approach are described on an introductory level. Then a mechanism of the stabilization of this model by quantum effects is suggested. In particular, a stability condition involving m , the interaction intensity, and the parameters of the crystalline field is given. It is independent of the temperature and is satisfied if m is small enough and/or the tunnelling frequency is big enough. It is shown that under this condition the infinite-volume correlation function decays exponentially; hence, no phase transitions can arise at all temperatures.

We consider the dual BBGKY hierarchy for quasiobservables of manyparticle systems as a basis of nonequilibrium statistical mechanics and give a complete description of the evolution of quasiobservables.

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Harmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L 2 -spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits, in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.