General contact processes: inhomogeneous models, models on graphs and on manifolds

TL;DR

This paper extends the contact process model to inhomogeneous settings on graphs, manifolds, and metric spaces, proving the existence of invariant measures in the critical regime under certain conditions.

Contribution

It introduces a general framework for contact processes on various spaces and establishes the existence of invariant measures in the critical regime.

Findings

Existence of a one-parameter set of invariant measures in the critical regime.

Applicable to contact processes on lattices, graphs, and manifolds.

Conditions involve separation of trajectories in the associated Markov jump process.

Abstract

The contact process is a particular case of birth-and-death processes on infinite particle configurations. We consider the contact models on locally compact separable metric spaces. We prove the existence of a one-parameter set of invariant measures in the critical regime under the condition imposed on the associated Markov jump process. This condition, roughly speaking, requires the separation of any pair of trajectories of this jump process. The general scheme can be applied to the contact process on the lattice in a heterogeneous and random environments as well as to the contact process on graphs and on manifolds.