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

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.