Binary jumps in continuum. I. Equilibrium processes and their scaling limits

Abstract

Let Γ denote the space of all locally finite subsets (configurations) in R d . A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over R d . In this paper, we study an equilibrium dynamics of binary jumps for which a Poisson measure is a symmetrizing (and hence invariant) measure. The existence and uniqueness of the corresponding stochastic dynamics are shown. We next prove the main result of this paper: a big class of dynamics of binary jumps converge, in a diffusive scaling limit, to a dynamics of interacting Brownian particles. We also study another scaling limit, which leads us to a spatial birth-and-death process in continuum. A remarkable property of the limiting dynamics is that its generator possesses a spectral gap, a property which is hopeless to expect from the initial dynamics of binary jumps.MSC: 60F99, 60J60, 60J75, 60K35

Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit

“…Example 4.8 (cf. [10]). Let c be given by (4.30) with c ′ (x 1 , x 2 , y 1 , y 2 ) = κa(x 1 − y 1 )a(x 2 − y 2 ) b(x 1 − x 2 ) + b(y 1 − y 2 ) .…”

“…A Poisson measure may be invariant, or even symmetrizing for such a dynamics. In the first part of the present paper [10], we considered such a process with generator…”