A quasispecies continuous contact model in a critical regime

TL;DR

This paper introduces a new non-equilibrium dynamical model in high-dimensional space, demonstrating the existence of a family of invariant measures and convergence properties, with a focus on critical regimes where particle density varies.

Contribution

It develops a marked continuous contact model in $d$-dimensional space and proves the existence of invariant measures and convergence in a critical regime, extending previous models.

Findings

Existence of a one-parameter family of invariant measures.

Convergence of the process to invariant measures from a marked Poisson measure.

Particle density is not conserved in this model.

Abstract

We study a new non-equilibrium dynamical model: a marked continuous contact model in $d$-dimensional space ($d \ge 3$). We prove that for certain values of rates (the critical regime) this system has the one-parameter family of invariant measures labelled by the spatial density of particles. Then we prove that the process starting from the marked Poisson measure converges to one of these invariant measures. In contrast with the continuous contact model studied earlier in \cite{KKP}, now the spatial particle density is not a conserved quantity.