Abstract
The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.
Highlights
The contact process is a stochastic model for the spread of an infection among the members of a population
An infected individual will infect each of its neighbors at rate λ > 0, and recover at rate 1. This evolution defines an interacting particle system whose state at time t is a subset ηt ⊆ Z, or equivalently an element ηt ∈ {0, 1}Z
In this paper we show that, despite non-existence of stationary measures, the subcritical contact process seen from the rightmost point does converge in distribution
Summary
The contact process is a stochastic model for the spread of an infection among the members of a population. Schonmann [14] showed that in this phase, planar oriented percolation seen from its rightmost point does not have any invariant measure on Σ0 This result was extended to the contact process by Andjel, Schinazi, and Schonmann [2]. In this paper we show that, despite non-existence of stationary measures, the subcritical contact process seen from the rightmost point does converge in distribution. The limiting measure is quasi-stationary and is supported on configurations that contain finitely many infected sites This extends an analogous result for subcritical planar oriented percolation [1]. The subcritical contact process seen from the rightmost point (ζt)t 0 has a unique minimal quasi-stationary distribution ν This measure ν is supported on finite configurations. Some of the main arguments in this paper come from the second author’s thesis [7]
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