Duality and Complete Convergence for Multi-type Additive Growth Models

TL;DR

This paper studies multi-type additive growth models, establishing duality, characterizations, and conditions for positive correlations, and proves complete convergence for several models including the two-stage contact process.

Contribution

It extends Harris's result by linking additivity to duality, provides new characterizations, and proves complete convergence for a broad class of models.

Findings

Additivity is equivalent to the existence of a dual process.

Necessary and sufficient conditions for positive correlations.

Complete convergence holds for models like the two-stage contact process.

Abstract

We consider a class of multi-type particle systems having similar structure to the contact process and show that additivity is equivalent to the existence of a dual process, extending a result of Harris. We give two additional characterizations of these systems, in spacetime as percolation models, and biologically as population models in which the interactions are due to crowding. We prove a necessary and sufficient condition for the model to preserve positive correlations. We then show that complete convergence on $\mathbb{Z}^d$ holds for a large subclass of models including the two-stage contact process and a household model, and give examples.