Non-equilibrium stochastic dynamics in continuum: The free case

Abstract

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

Laplace operators on the cone of Radon measures

“…(iii) This statement can be easily derived from Corollary 20 analogously to the proof of Theorem 5.1 in [16], see also the proof of Theorem 2.2 in [17].…”

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“…This can only hold under additional conditions on the initial configuration, for details, see Ref. 10.…”

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“…The operator L 0 , realized in the Fock space F (L 2 (R d , z dx)), is the differential second quantization of the operator a b 1, so the corresponding dynamics is 'free', i.e., without interaction between particles, see [18,28,29] for further detail. E 0 (F, F ) ≥ a 2 z b Γ (F (γ) − F πz ) 2 π z (dγ), F ∈ D(E 0 ), (5.17) where F πz := Γ F (γ)π z (dγ).…”