Functional central limit theorem for the interface of the multitype contact process

TL;DR

This paper proves that the interface between two species in a multitype contact process on the integer lattice converges to Brownian motion under diffusive scaling, providing a functional central limit theorem for the process.

Contribution

It establishes a functional central limit theorem for the interface of the multitype contact process, showing convergence to Brownian motion.

Findings

Interface position converges to Brownian motion under diffusive scaling

Provides rigorous proof of the central limit behavior for the interface

Extends understanding of spatial stochastic processes in ecology and epidemiology

Abstract

We study the interface of the multitype contact process on $\mathbb{Z}$. In this process, each site of $\mathbb{Z}$ is either empty or occupied by an individual of one of two species. Each individual dies with rate 1 and attempts to give birth with rate $2 R \lambda$; the position for the possible new individual is chosen uniformly at random within distance $R$ of the parent, and the birth is suppressed if this position is already occupied. We consider the process started from the configuration in which all sites to the left of the origin are occupied by one of the species and all sites to the right of the origin by the other species, and study the evolution of the region of interface between the two species. We prove that, under diffusive scaling, the position of the interface converges to Brownian motion.