Existence of a stationary distribution for multi-dimensional infinite volume forest-fire processes

TL;DR

This paper proves the existence of a stationary distribution for multi-dimensional infinite volume forest-fire processes on Z^d, establishing a foundational result for understanding long-term behavior in such stochastic models.

Contribution

It demonstrates the existence of a stationary distribution for forest-fire processes on Z^d by constructing it as a limit of finite volume invariant distributions, a novel approach in this context.

Findings

Stationary distribution exists for forest-fire processes on Z^d, d>=2

Constructed as a limit of finite volume invariant distributions

Provides a basis for analyzing long-term dynamics of forest-fire models

Abstract

Consider the following forest-fire process on a connected graph. Each site of the graph can be either occupied or vacant. A vacant site becomes occupied with rate 1. A site is ignited with rate lambda, and its whole occupied cluster burns instantaneously. The purpose of this paper is to show the existence of a stationary distribution for forest-fire processes on Z^d, for d >= 2. We define a distribution as a limit of a sequence of invariant distributions of finite volume forest-fire processes, and then show it is a stationary distribution for forest-fire processes on Z^d, d>=2.