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

• Published in: Journal of mathematical physics 2011-06, Vol.52 (6), p.063304-063304-25
• Main Authors: Finkelshtein, Dmitri L., Kondratiev, Yuri G., Kutoviy, Oleksandr V., Lytvynov, Eugene
• Format: Article
• Language: English
• Subjects: Brownian motion; Exact sciences and technology; Markov analysis; Mathematical methods in physics; Mathematics; Physics; Poisson distribution; Sciences and techniques of general use; Stochastic models
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.3601118

Let Γ denote the space of all locally finite subsets (configurations) in \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^d$\end{document} R d . 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} 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.