Voting models and semilinear parabolic equations

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 f ( 0 ) = f ( 1 ) = 0 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.