Source: Ann. Inst. H. Poincaré Probab. Statist. Volume 48, Number 3
(2012), 661-687.
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 pathwise explanation 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.
Nous proposons une approche probabiliste au problème classique de l’existence, de l’unicité et du comportement asymptotique des solutions monotones de l’équation de propagation de front associée à l’équation parabolique du super-mouvement brownien de mécanisme de branchement général. Bien que largement inspiré par l’approche de Kyprianou (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53–72) pour le mouvement brownien branchant, cet article ouvre plusieurs perspectives nouvelles. Notre analyse inclut le rôle de la normalisation de Seneta–Heyde qui, dans cette situation, s’inspire du travail classique de Grey (J. Appl. Probab. 11 (1974) 669–677). Nous donnons une explication trajectorielle de la décomposition en épine (la particule immortelle d’Evans), en utilisant la $\mathbb{N}$-mesure de Dynkin–Kuznetsov comme ingrédient clef. En outre, dans l’esprit des lignes d’arrêt de Neveu nous utilisons à plusieurs reprises les mesures de sortie de Dynkin. La nature générale du mécanisme de branchement rend l’analyse du problème plus délicate et nous proposons une dichotomie exacte basée sur un moment $X(\log X)^{2}$ pour la convergence presque-sûre de la martingale dérivée (pour la valeur critique de son paramètre) vers une limite non-triviale. Ceci diffère du cas du mouvement brownien branchant (Ann. Inst. Henri Poincaré Probab. Stat. 40 (2004) 53–72) et de la marche aléatoire branchante (Adv. in Appl. Probab. 36 (2004) 544–581), où un écart dans les hypothèses sur les moments apparaît entre les conditions nécessaires et suffisantes. Notre approche probabiliste permet de retrouver des résultats connus d’existence, d’unicité et de comportement asymptotique pour l’équation de propagation de front reliée au super-mouvement brownien.
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