数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (5): 1345-1362.doi: 10.1007/s10473-019-0513-y

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ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

王月娇1, 刘再明2, 刘全升3,4, 李应求3,4   

  1. 1. College of Mathematics and Computational Science, Hunan First Normal University, Changsha 410205, China;
    2. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
    3. LMBA, UMR CNRS 6205, Université de Bretagne-Sud, F-56000 Vannes, France;
    4. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410004, China
  • 收稿日期:2017-11-01 修回日期:2019-05-17 出版日期:2019-10-25 发布日期:2019-11-11
  • 通讯作者: Quansheng LIU E-mail:quansheng.liu@univ-ubs.fr
  • 作者简介:Yuejiao WANG,E-mail:wangyuejiaohujing@163.com;Zaiming LIU,E-mail:math_lzm@csu.edu.cn;Yingqiu LI,E-mail:liyq-2001@163.com
  • 基金资助:
    The work has benefited from the support of the French government Investissements d'Avenir program ANR-11-LABX-0020-01. It has been partially supported by the National Natural Science Foundation of China (11571052, 11401590, 11731012 and 11671404), and by Hunan Natural Science Foundation (2017JJ2271).

ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME

Yuejiao WANG1, Zaiming LIU2, Quansheng LIU3,4, Yingqiu LI3,4   

  1. 1. College of Mathematics and Computational Science, Hunan First Normal University, Changsha 410205, China;
    2. School of Mathematics and Statistics, Central South University, Changsha 410083, China;
    3. LMBA, UMR CNRS 6205, Université de Bretagne-Sud, F-56000 Vannes, France;
    4. School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410004, China
  • Received:2017-11-01 Revised:2019-05-17 Online:2019-10-25 Published:2019-11-11
  • Contact: Quansheng LIU E-mail:quansheng.liu@univ-ubs.fr
  • Supported by:
    The work has benefited from the support of the French government Investissements d'Avenir program ANR-11-LABX-0020-01. It has been partially supported by the National Natural Science Foundation of China (11571052, 11401590, 11731012 and 11671404), and by Hunan Natural Science Foundation (2017JJ2271).

摘要: 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), with Zn denoting the counting measure of particles of generation n, and Eξ 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.

关键词: branching random walk, random environment, quenched moments, weighted moments

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), with Zn denoting the counting measure of particles of generation n, and Eξ 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.

Key words: branching random walk, random environment, quenched moments, weighted moments

中图分类号: 

  • 60K37