Waves Everywhere: A Distributional Equation Approach to Front Propagation

Abstract

We study reaction-diffusion particle systems with several interaction mechanisms. As the number of particles tends to infinity, the system admits a mean-field limit describing the bulk behaviour. We focus on determining the propagation speed and the particle distribution around the centre of mass, which corresponds to the travelling wave of the limiting equation. We introduce a probabilistic method to characterise these waves via tagged particle distributional equations. Our key technique connects these to linear distributional equations solvable using martingale limits from branching processes. We first demonstrate our approach on a general model where particles move via L\'evy processes and synchronise at interaction moments (the lower particle jumps to the position of the higher one). Assuming the mean-field limit holds, we characterise its travelling-wave solutions. We then apply the method to two specific models with established mean-field limits. For Brownian particles, we recover known travelling-wave solutions of the F-KPP equation. For the compound Poisson model studied in \cite{baryshnikov2025large} where particles perform random walks with exponential holding times and copy positions at interactions, we extend previous results beyond exponential jumps to arbitrary jump distributions. Finally, we analyse the power-of-2 growth model, where interactions add a random value to the lower particle. We characterise its travelling waves and discover a surprising connection to the synchronisation models. The distributional equations at the heart of our technique are of independent interest, and we identify connections and differences with related equations studied extensively in the literature.