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2014 Subcritical contact processes seen from a typical infected site
Anja Sturm, Jan Swart
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Electron. J. Probab. 19: 1-46 (2014). DOI: 10.1214/EJP.v19-2904


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.


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Anja Sturm. Jan Swart. "Subcritical contact processes seen from a typical infected site." Electron. J. Probab. 19 1 - 46, 2014.


Accepted: 23 June 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1294.82030
MathSciNet: MR3227062
Digital Object Identifier: 10.1214/EJP.v19-2904

Primary: 82C22
Secondary: 60K35 , 82B43

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

Vol.19 • 2014
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