## Electronic Journal of Probability

### Subcritical contact processes seen from a typical infected site

#### Abstract

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.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 53, 46 pp.

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

https://projecteuclid.org/euclid.ejp/1465065695

Digital Object Identifier
doi:10.1214/EJP.v19-2904

Mathematical Reviews number (MathSciNet)
MR3227062

Zentralblatt MATH identifier
1294.82030

Rights

#### Citation

Sturm, Anja; Swart, Jan. Subcritical contact processes seen from a typical infected site. Electron. J. Probab. 19 (2014), paper no. 53, 46 pp. doi:10.1214/EJP.v19-2904. https://projecteuclid.org/euclid.ejp/1465065695

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