A Widom–Rowlinson Jump Dynamics in the Continuum

Abstract

We study the dynamics of an infinite system of point particles of two types. They perform random jumps in R d in the course of which particles of different types repel each other whereas those of the same type do not interact. The states of the system are probability measures on the corresponding configuration space. The main result is the construction of the global (in time) Markov evolution of such states by means of correlation functions. It is proved that for each initial sub-Poissonian state μ 0 , the states evolve μ 0 → μ t in such a way that μ t is sub-Poissonian for all t > 0. The mesoscopic (approximate) description of the evolution of states is also given. The stability of translation invariant stationary states is studied. In particular, we show that some of such states can be unstable with respect to space-dependent perturbations.

Evolution of infinite populations of immigrants: Micro- and mesoscopic description

“…where ω ⊂ R d is the configuration of some entities (attraction centers), distributed e.g., according to a Poisson law, and Φ(x, y) is an attraction/repulsion potential. Another possibility, close to just mentioned, is to consider a two-type system of the Widom-Rowlinson type [3,8,15,16]. In this system, the particles of different types would repel each other whereas those of the same type not interact.…”

“…where ω ± are as in (2.40 Here we construct the evolution q 0,ε → q t,ε mentioned in the theorem. For a conceptual background of this approach, see the corresponding parts of [3,7] and the references therein. Let L ∆ ε be the operator defined in (2.27) in which φ is multiplied by ε ∈ (0, 1].…”

Evolution of States of a Continuum Jump Model with Attraction

“…An example can be [7], that k t is a correlation function allowed there for continuing to all t > 0 the solution primarily obtained on a bounded time interval. For jump dynamics with repulsion, such continuation was realized in [3,4], also by means of the corresponding property of k t . However, for the model considered here for such a continuation to be done proving that the solution k t is a correlation function -and hence is positive in a certain sense -might not be enough.…”

A Markov process for an infinite interacting particle system in the continuum

“…In [5], there was studied a model in which point particles of two types perform random jumps over R d . Their common dynamics are described by the corresponding analog of the Kolmogorov operator (1.2) in which particles of different types repel each other, whereas those of the same type do not interact.…”