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

TL;DR

This paper investigates equilibrium binary jump processes in continuum, proving their existence, uniqueness, and showing they converge to interacting Brownian motions or birth-death processes under scaling, with notable spectral gap properties.

Contribution

It establishes the existence and uniqueness of equilibrium binary jump dynamics and demonstrates their convergence to diffusive and birth-death processes in continuum.

Findings

Binary jump dynamics are well-defined and unique.

Scaling limits lead to interacting Brownian motions.

Spectral gap property appears in the limiting dynamics.

Abstract

Let $\Gamma$ 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 $\Gamma$ 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.