Supercritical super-Brownian motion with a general branching mechanism and travelling waves

TL;DR

This paper studies the existence, uniqueness, and asymptotics of traveling wave solutions for super-Brownian motion with a general branching mechanism, extending probabilistic methods and providing new insights into the behavior of associated martingales.

Contribution

It introduces a pathwise explanation of the spine decomposition using Dynkin-Kuznetsov measures and establishes an exact moment dichotomy for derivative martingale convergence.

Findings

Pathwise explanation of the spine decomposition.

Exact X(log X)^2 moment dichotomy for derivative martingale.

Extension of probabilistic methods to general branching mechanisms.

Abstract

We consider the classical problem of existence, uniqueness and asymptotics of monotone solutions to the travelling wave equation associated to the parabolic semi-group equation of a super-Brownian motion with a general branching mechanism. Whilst we are strongly guided by the probabilistic reasoning of Kyprianou (2004) for branching Brownian motion, the current paper offers a number of new insights. Our analysis incorporates the role of Seneta-Heyde norming which, in the current setting, draws on classical work of Grey (1974). We give a pathwise explanation of Evans' immortal particle picture (the spine decomposition) which uses the Dynkin-Kuznetsov N-measure as a key ingredient. Moreover, in the spirit of Neveu's stopping lines we make repeated use of Dynkin's exit measures. Additional complications arise from the general nature of the branching mechanism. As a consequence of the analysis we also offer an exact X(log X)^2 moment dichotomy for the almost sure convergence of the so-called derivative martingale at its critical parameter to a non-trivial limit. This differs to the case of branching Brownian motion and branching random walk where a moment `gap' appears in the necessary and sufficient conditions.