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

Abstract

Let Γ denote the space of all locally finite subsets (configurations) in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd. A stochastic dynamics of binary jumps in continuum is a Markov process on Γ in which pairs of particles simultaneously hop over \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document}Rd. 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.