Applying monoid duality to a double contact process

Abstract

In this paper we use duality techniques to study a coupling of the well-known contact process (CP) and the 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 components, 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 parameters 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 (modeling “infected individuals”) can only be present in the cCP 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.