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

We oer a probabilistic treatment of 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 reasoning in Kyprianou [26] for branching Brownian motion, the current paper oers 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 [20]. 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 oer an exact X(logX) 2 moment dichotomy for the almost sure convergence of the socalled derivative martingale at its critical parameter to a non-trivial limit. This diers to the case of branching Brownian motion, [26], and branching random walk, [2], where a moment ‘gap’ appears in the necessary and sucient conditions. Our probabilistic treatment allows us to replicate known existence, uniqueness and asymptotic results for the travelling wave equation, which is related to a super-Brownian motion.

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