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Journal Article

Subcritical contact processes seen from a typical infected site

Sturm A., Swart Jan M.

: Electronic Journal of Probability vol.19, 1 (2014)

: GA201/09/1931, GA ČR, GAP201/12/2613, GA ČR

: contact process, exponential growth rate, eigenmeasure, Campbell law, Palm law, quasi-invariant law

: 10.1214/EJP.v19-2904


(eng): 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.

: BA