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

TL;DR

This paper investigates the scaling limits and fluctuations of a family of $N$-urn branching processes, revealing their behavior through differential equations and Ornstein-Uhlenbeck processes, with applications to hitting time limit theorems.

Contribution

It introduces a unified framework for analyzing $N$-urn branching processes, including the Ehrenfest model and branching random walk, using differential equations and stochastic processes.

Findings

Scaling limit described by a linear ODE in $C(\

\mathbb{T})$

Fluctuations driven by a generalized Ornstein-Uhlenbeck process in the dual space of $C^\\infty(\mathbb{T})$

Abstract

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