Spines, skeletons and the strong law of large numbers for superdiffusions
type: Item
instance of: scholarly article
title: Spines, skeletons and the strong law of large numbers for superdiffusions
zbMATH Open document ID: 1330.60052
DOI: 10.1214/14-AOP944
author: Maren Eckhoff
author: Matthias Winkel
author: A. E. Kyprianou
published in: The Annals of Probability
publication date: 2015-10-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1309.6196
full work available at URL: https://projecteuclid.org/euclid.aop/1441792293
review text: The paper investigates almost sure limit behavior of a supercritical superdiffusion \({({X_t})_{t \geqslant 0}}\) on a domain \(D \subseteq {{\mathbb R}^d}\) with branching mechanism \[ (x,z) \mapsto - \beta (x) + \alpha (x){z^2} + \int_{(0,\infty )} {({e^{ - zy}} - 1 + zy)\Pi (x,dy)}. \] It is shown that for suitable test functions \(f\) and starting measures \(\mu \), \[ \frac{{\left\langle {f,{X_t}} \right\rangle }}{{{P_\mu }\left[ {\left\langle {f,{X_t}} \right\rangle } \right]}} \to {W_\infty } \] \({P_\mu }\)-almost surely as \(t \to \infty \), where \({W_\infty }\) is a finite, non-deterministic random variable characterized as a martingale limit. The authors use the skeleton decomposition that represents the superprocess as an immigration process along a branching particle process (the skeleton), where immigration occurs in a Poisson way along the space-time trajectories and at the branch points of the skeleton.
reviewed by: Oleg K. Zakusilo
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60J68
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 6502682
zbMATH Keywords: superdiffusions
zbMATH Keywords: measure-valued diffusions
zbMATH Keywords: strong law of large numbers
zbMATH Keywords: almost sure limit theorem
zbMATH Keywords: branching mechanism
zbMATH Keywords: skeleton decomposition
zbMATH Keywords: spine decomposition
zbMATH Keywords: martingales
MaRDI profile type: Publication
arXiv ID: 1309.6196
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Spines, skeletons and the strong law of large numbers for superdiffusions (OpenAlex ID)
OpenAlex ID: W2168621054