A quasispecies continuous contact model in a subcritical regime

TL;DR

This paper analyzes a spatial contact process with spontaneous births in a subcritical regime, proving the existence of an invariant measure and convergence of the process to this measure from any initial state.

Contribution

It introduces a subcritical contact model with external spontaneous births and establishes the existence and uniqueness of its invariant measure.

Findings

Existence of an invariant measure for the model.

Convergence of the process to the invariant measure from any initial distribution.

Abstract

We study a non-equilibrium dynamical model: a marked continuous contact model in $d$-dimensional space, $d \ge 1$. In contrast with the continuous contact model in a critical regime, see \cite{KKP}, \cite{KPZ}, the model under consideration is in the subcritical regime and it contains an additional spontaneous spatially homogeneous birth from an external source. We prove that this system has an invariant measure. We prove also that the process starting from any initial distribution converges to this invariant measure.