Supercritical super-Brownian motion with a general branching mechanism and travelling waves
type: Item
instance of: scholarly article
title: Supercritical super-Brownian motion with a general branching mechanism and travelling waves
zbMATH Open document ID: 1267.60094
DOI: 10.1214/11-AIHP448
author: A. E. Kyprianou
author: R.-L. Liu
author: A. Murillo-Salas
author: Y.-X. Ren
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2012-08-20T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.3659
full work available at URL: https://projecteuclid.org/euclid.aihp/1340714867
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 60J68
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60E10
zbMATH DE Number: 6069309
zbMATH Keywords: superprocess
zbMATH Keywords: spin decomposition
zbMATH Keywords: additive martingale
zbMATH Keywords: derivative martingale
zbMATH Keywords: travelling wave
zbMATH Keywords: supercriticality
zbMATH Keywords: general branching mechanism
MaRDI profile type: Publication
arXiv ID: 1005.3659
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review text: This paper treats, in a probabilistic manner, the classical problem of existence, uniqueness and asymptotic behaviour of monotone solutions to the travelling wave equation associated with the parabolic semigroup equation of a super-Brownian motion \(X_t\) with a general branching mechanism \(\psi\). The leading philosophy is strongly due to the work done by the first author's [Ann. Inst. Henri Poincaré, Probab. Stat. 40, No. 1, 53--72 (2004; Zbl 1042.60057)] for branching Brownian motions and the travelling wave solution to the KPP equation.NEWLINENEWLINEMore precisely, let \(X = \{ X_t ; t \geq 0 \}\) denote a one-dimensional \(\psi\)-super-Brownian motion with general branching mechanism NEWLINE\[CARRIAGE_RETURNNEWLINE \psi(\lambda) = - \alpha \lambda + \beta \lambda^2 + \int_{(0, \infty)} ( e^{- \lambda x} - 1 + \lambda x) \nu(dx) \quad \text{for } \lambda \geq 0, \tag{1} CARRIAGE_RETURNNEWLINE\]NEWLINE where \(\alpha = - \psi'(0+) \in (0, \infty)\), \(\beta \geq 0\) and \(\nu\) is a measure concentrated on \((0, \infty)\) satisfying \(\int_{(0, \infty)} ( x \wedge x^2) \nu(dx)<\infty\), and let \({\mathcal M}_F({\mathbb R})\) be the space of finite measures on \({\mathbb R}\). Note that \(X\) is an \({\mathcal M}_F({\mathbb R})\)-valued Markov process under \(\operatorname{P}_{\mu}\), where \(\operatorname{P}_{\mu}\) is the law of \(X\) with initial configuration \(\mu \in {\mathcal M}_F({\mathbb R})\). For \(f \in C_b^+({\mathbb R})\) (the space of finite, uniformly bounded, continuous functions on \({\mathbb R}\)) and \(\mu \in {\mathcal M}_F({\mathbb R})\), define \(\langle f, \mu \rangle=\int_{{\mathbb R}} f(x) \mu(dx)\), and accordingly write \(\| \mu \|=\langle 1, \mu \rangle\). The existence of the superprocess \(X_t\) is guaranteed by results of \textit{E. B. Dynkin} [Ann. Probab. 21, No. 3, 1185--1262 (1993; Zbl 0806.60066)]. The characterization of the superprocess \(X_t\) is given by the following. For all \(f \in C_b^+({\mathbb R})\) and \(\mu \in {\…
description: scientific article; zbMATH DE number 6069309