Voting models and semilinear parabolic equations

Abstract

We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975 Commun. Pure Appl. Math. 28 323–31). In particular, we present ‘random outcome’ and ‘random threshold’ voting models that yield any polynomial nonlinearity f satisfying and a ‘recursive up the tree’ model that allows to go beyond this restriction on f. We compute several examples of particular interest; for example, we obtain a curious interpretation of the heat equation in terms of a nontrivial voting model and a ‘group-based’ voting rule that leads to a probabilistic view of the pushed-pulled transition for a class of nonlinearities introduced by Ebert and van Saarloos.