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

A. E. Kyprianou; R.-L. Liu; A. Murillo-Salas; Y.-X. Ren

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 3, page 661-687
  • ISSN: 0246-0203

Abstract

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We 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 pathwiseexplanation 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 (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.

How to cite

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Kyprianou, A. E., et al. "Supercritical super-brownian motion with a general branching mechanism and travelling waves." Annales de l'I.H.P. Probabilités et statistiques 48.3 (2012): 661-687. <http://eudml.org/doc/271940>.

@article{Kyprianou2012,
abstract = {We 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 pathwiseexplanation of Evans’ immortal particle picture (the spine decomposition) which uses the Dynkin–Kuznetsov $\mathbb \{N\}$-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 (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.},
author = {Kyprianou, A. E., Liu, R.-L., Murillo-Salas, A., Ren, Y.-X.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {superprocesses; $\mathbb \{N\}$-measure; spine decomposition; additive martingale; derivative martingale; travelling waves; superprocess; spin decomposition; additive martingale; derivative martingale; travelling wave; supercriticality; general branching mechanism},
language = {eng},
number = {3},
pages = {661-687},
publisher = {Gauthier-Villars},
title = {Supercritical super-brownian motion with a general branching mechanism and travelling waves},
url = {http://eudml.org/doc/271940},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Kyprianou, A. E.
AU - Liu, R.-L.
AU - Murillo-Salas, A.
AU - Ren, Y.-X.
TI - Supercritical super-brownian motion with a general branching mechanism and travelling waves
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 3
SP - 661
EP - 687
AB - We 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 pathwiseexplanation of Evans’ immortal particle picture (the spine decomposition) which uses the Dynkin–Kuznetsov $\mathbb {N}$-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 (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.
LA - eng
KW - superprocesses; $\mathbb {N}$-measure; spine decomposition; additive martingale; derivative martingale; travelling waves; superprocess; spin decomposition; additive martingale; derivative martingale; travelling wave; supercriticality; general branching mechanism
UR - http://eudml.org/doc/271940
ER -

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