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Asymptotic Properties of a Branching Random Walk with a Random Environment in Time

Authors
  • Wang, Yuejiao1
  • Liu, Zaiming2
  • Liu, Quansheng3, 4
  • Li, Yingqiu4
  • 1 Hunan First Normal University, College of Mathematics and Computational Science, Changsha, 410205, China , Changsha (China)
  • 2 Central South University, School of Mathematics and Statistics, Changsha, 410083, China , Changsha (China)
  • 3 Université de Bretagne-Sud, LMBA, UMR CNRS 6205, Vannes, F-56000, France , Vannes (France)
  • 4 Changsha University of Science and Technology, School of Mathematics and Statistics, Changsha, 410004, China , Changsha (China)
Type
Published Article
Journal
Acta Mathematica Scientia
Publisher
Springer-Verlag
Publication Date
Jul 10, 2019
Volume
39
Issue
5
Pages
1345–1362
Identifiers
DOI: 10.1007/s10473-019-0513-y
Source
Springer Nature
Keywords
License
Yellow

Abstract

We consider a branching random walk in an independent and identically distributed random environment ξ = (ξn) indexed by the time. Let W be the limit of the martingale Wn=∫e−txZn(dx)/Eξ∫e−txZn(dx)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_n=\int\;e^{-tx}Z_n(\text{d}x)/\mathbb{E}_\xi\int\;e^{-tx}Z_n(\text{d}x)$$\end{document}, with Zn denoting the counting measure of particles of generation n, and Eξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{E}_\xi$$\end{document} the conditional expectation given the environment ξ. We find necessary and sufficient conditions for the existence of quenched moments and weighted moments of W, when W is non-degenerate.

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