Subcritical contact processes seen from a typical infected site

摘要

What is the long-time behavior of the law of a contact process started?with a single infected site, distributed according to counting measure?on the lattice? This question is related to the configuration as seen?from a typical infected site and gives rise to the definition of?so-called eigenmeasures, which are possibly infinite measures on the?set of nonempty configurations that are preserved under the dynamics?up to a time-dependent exponential factor. In this paper, we study?eigenmeasures of contact processes on general countable groups in the?subcritical regime. We prove that in this regime, the process has a?unique spatially homogeneous eigenmeasure. As an application, we show?that the law of the process as seen from a typical infected site,?chosen according to a Campbell law, converges to a long-time limit. We?also show that the exponential decay rate of the expected number of?infected sites is continuously differentiable and strictly increasing?as a function of the recovery rate, and we give a formula for the?derivative in terms of the long time limit law of the process as seen?from a typical infected site.