%0 Generic %1 kyprianou2010supercritical %A Kyprianou, A. E. %A Liu, R. L. %A Murillo-Salas, A. %A Ren, Y. X. %D 2010 %K branching_Brownian_motion superprocesses travelling_wave %T Supercritical super-Brownian motion with a general branching mechanism and travelling waves %U http://arxiv.org/abs/1005.3659 %X 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.

@misc{kyprianou2010supercritical, 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. }, added-at = {2010-06-16T00:24:13.000+0200}, author = {Kyprianou, A. E. and Liu, R. L. and Murillo-Salas, A. and Ren, Y. X.}, biburl = {https://www.bibsonomy.org/bibtex/2307199a07791fc18b70b6f5fcc8ad37b/peter.ralph}, description = {[1005.3659] Supercritical super-Brownian motion with a general branching mechanism and travelling waves}, interhash = {85de9d9f4c8545fea87a9eb9a8efb458}, intrahash = {307199a07791fc18b70b6f5fcc8ad37b}, keywords = {branching_Brownian_motion superprocesses travelling_wave}, note = {cite arxiv:1005.3659 Comment: 31 pages}, timestamp = {2013-09-12T22:23:01.000+0200}, title = {Supercritical super-Brownian motion with a general branching mechanism and travelling waves}, url = {http://arxiv.org/abs/1005.3659}, year = 2010 }