Scaling limits and fluctuations of a family of N-urn branching processes

Abstract

In this paper, we are concerned with a family of N-urn branching processes, where some particles are initially placed in N urns, and then each particle gives birth to several new particles in some urn when dies. This model includes the N-urn Ehrenfest model and N-urn branching random walk as special cases. We show that the scaling limit of the process is driven by a C(T)-valued linear ordinary differential equation and the fluctuation of the process is driven by a generalized Ornstein–Uhlenbeck process in the dual of C∞(T), where T=(0,1] is the one-dimensional torus. A crucial step for the proofs of the above main results is to show that numbers of particles in different urns are approximately independent. As applications of our main results, the limit theorems of the hitting times of the process are also discussed.