Research

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


Reference:

Kyprianou, A. E., Liu, R. L., Murillo-Salas, A. and Ren, Y. X., 2012. Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Annales De L Institut Henri Poincare-Probabilites Et Statistiques, 48 (3), pp. 661-687.

Related documents:

[img]
Preview
PDF (Kyprianou_AIHPPS_2012_48_3_661.pdf) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (336kB) | Preview

    Official URL:

    http://dx.doi.org/10.1214/11-AIHP448

    Abstract

    e offer 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 (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53–72) 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 (J. Appl. Probab. 11 (1974) 669–677). 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(logX)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 (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53–72), and branching random walk (Adv. in Appl. Probab. 36 (2004) 544–581), where a moment ‘gap’ appears in the necessary and sufficient 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.

    Details

    Item Type Articles
    CreatorsKyprianou, A. E., Liu, R. L., Murillo-Salas, A. and Ren, Y. X.
    DOI10.1214/11-AIHP448
    DepartmentsFaculty of Science > Mathematical Sciences
    RefereedYes
    StatusPublished
    ID Code30463

    Export

    Actions (login required)

    View Item

    Document Downloads

    More statistics for this item...