Hydrodynamics of a class of $N$-urn linear systems

TL;DR

This paper investigates the hydrodynamic behavior and fluctuations of a broad class of $N$-urn linear systems, including models like voter models and exclusion processes, revealing their limits are governed by linear ODEs and Ornstein-Uhlenbeck processes.

Contribution

It establishes the hydrodynamic limit and fluctuation results for a general class of $N$-urn linear systems, extending previous models with new analytical techniques.

Findings

Hydrodynamic limit described by a linear ODE in the dual space

Fluctuations characterized by a dual Ornstein-Uhlenbeck process

Key role of an extended Chapman-Kolmogorov equation in proofs

Abstract

In this paper we are concerned with hydrodynamics of a class of $N$-urn linear systems, which include voter models, pair-symmetric exclusion processes and binary contact path processes on $N$ urns as special cases. We show that the hydrodynamic limit of our process is driven by a $\left(C[0,1]\right)^\prime$-valued linear ordinary differential equation and the fluctuation of our process, i.e, central limit theorem from the hydrodynamic limit, is driven by a $\left(C[0, 1]\right)^\prime$-valued Ornstein-Uhlenbeck process. To derive above main results, we need several replacement lemmas. An extension in linear systems of Chapman-Kolmogorov equation plays key role in proofs of these replacement lemmas.