Voting models and semilinear parabolic equations

TL;DR

This paper develops probabilistic voting models on genealogical trees of branching Brownian motion to interpret solutions of semi-linear parabolic equations with polynomial nonlinearities, extending classical connections like Fisher-KPP.

Contribution

It introduces new voting models that represent polynomial nonlinearities and a recursive model to handle more general functions, broadening the probabilistic interpretation of these equations.

Findings

Probabilistic voting models correspond to solutions of semi-linear parabolic equations.

New models interpret heat equation and nonlinearities related to biological transitions.

Extended the classical McKean connection to more general polynomial nonlinearities.

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 the connection between the Fisher-KPP equation and BBM discovered by McKean in~\cite{McK}. 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 a few 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 pushmi-pullyu transition for a class of nonlinearities introduced by Ebert and van Saarloos.