The contact process as seen from a random walk

TL;DR

This paper studies a random walk on a contact process in high infection or low jump rate regimes, showing convergence to extremal measures and establishing a law of large numbers for the walk.

Contribution

It introduces the analysis of the contact process as seen from a random walk, extending known results to new regimes and providing a law of large numbers.

Findings

At most two extremal measures for the process under certain conditions

Characterization of convergence based on contact process survival

Law of large numbers for the random walk

Abstract

We consider a random walk on top of the contact process on $\mathbb{Z}^d$ with $d\geq 1$. In particular, we focus on the "contact process as seen from the random walk". Under the assumption that the infection rate of the contact process is large or the jump rate of the random walk is small, we show that this process has at most two extremal measures. Moreover, the convergence to these extremal measures is characterised by whether the contact process survives or dies out, similar to the complete convergence theorem known for the ordinary contact process. Using this, we furthermore provide a law of large numbers for the random walk which holds under general assumptions on the jump probabilities of the random walk.