A stochastic model for competing growth on $\mathbb{R}^d$

Abstract

A stochastic model, describing the growth of two competing infections on $\mathbb{R}^d$, is introduced. The growth is driven by outbursts in the infected region, an outburst in the type 1 (2) infected region transmitting the type 1 (2) infection to the previously uninfected parts of a ball with stochastic radius around the outburst point. The main result is that with the growth rate for one of the infection types fixed, mutual unbounded growth has probability zero for all but at most countably many values of the other infection rate. This is a continuum analog of a result of H\"{a}ggstr\"{o}m and Pemantle. We also extend a shape theorem of Deijfen for the corresponding model with just one type of infection.