Applying monoid duality to a double contact process

TL;DR

This paper employs duality techniques to analyze a combined contact process and annihilating branching process, establishing conditions for unique invariant laws and extending duality methods for infinite state space particle systems.

Contribution

It introduces the double contact process (2CP), combining CP and cCP, and extends duality theory to infinite state space interacting particle systems.

Findings

At most one invariant law exists for the 2CP.

Monotone coupling of CP and cCP is possible under certain parameters.

Extended dualities facilitate analysis of infinite state space processes.

Abstract

In this paper we use duality techniques to study a combination of the well-known contact process (CP) and the somewhat less-known annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two entries, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameter in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones in the cCP can only be present at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper "Commutative monoid duality" to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.