Asymptotic shape and the speed of propagation of continuous-time continuous-space birth processes

TL;DR

This paper establishes a shape theorem for a continuous-time, continuous-space stochastic growth model, providing insights into its asymptotic shape and propagation speed using subadditive ergodic theory.

Contribution

It extends classical lattice growth models to continuous space and time, offering a rigorous proof of shape and speed of propagation under general conditions.

Findings

Proves a shape theorem for the model

Provides a formula for propagation speed with truncated birth rate

Uses subadditive ergodic theorem for the proof

Abstract

We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similarly to the classical lattice growth models the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free branching birth rate.