## The Annals of Probability

### Edge Fluctuations for the One Dimensional Supercritical Contact Process

#### Abstract

We consider the one dimensional supercritical contact process with initial configurations having infinitely many particles to the left of the origin and only finitely many to its right. Starting from any such configuration, we first prove that in the limit as time goes to infinity the law of the process, as seen from the edge, converges to the invariant distribution constructed by Durrett [12]. We then prove a functional central limit theorem for the fluctuations of the edge around its average, showing that the corresponding diffusion coefficient is strictly positive. We finally characterize the space time structure of the system. In particular we prove that its distribution shifted in space by $\alpha t$ ($t$ denotes the time and $\alpha$ the drift of the edge) converges when $t$ goes to infinity to a $\frac{1}{2} - \frac{1}{2}$ mixture of the two extremal invariant measures for the contact process.

#### Article information

Source
Ann. Probab., Volume 15, Number 3 (1987), 1131-1145.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176992086

Digital Object Identifier
doi:10.1214/aop/1176992086

Mathematical Reviews number (MathSciNet)
MR893919

Zentralblatt MATH identifier
0645.60103

JSTOR
links.jstor.org

#### Citation

Galves, Antonio; Presutti, Errico. Edge Fluctuations for the One Dimensional Supercritical Contact Process. Ann. Probab. 15 (1987), no. 3, 1131--1145. doi:10.1214/aop/1176992086. https://projecteuclid.org/euclid.aop/1176992086