Invariant measures for spatial contact model in small dimensions

TL;DR

This paper investigates invariant measures of a continuous contact model in one and two dimensions, demonstrating the existence of a parameterized family of invariant measures and convergence from broad initial states.

Contribution

It establishes the existence of a one-parameter family of invariant measures for the model in small dimensions and proves convergence from broad initial conditions.

Findings

Existence of a one-parameter set of invariant measures in $d=1,2$

Convergence of the system to these invariant measures from broad initial states

Characterization of invariant measures in the critical regime

Abstract

We study invariant measures of continuous contact model in small dimensional spaces ($d =1,2$). Under general conditions we prove that in the critical regime this system has the one-parameter set of invariant measures parametrized by the spatial density of particles. Also for broad class of initial states we prove the convergence to one of these invariant measures.