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

doi:10.1007/s10473-019-0513-y

Acta mathematica scientia,Series B ›› 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

Yuejiao WANG¹, Zaiming LIU², Quansheng LIU^3,4, Yingqiu LI^3,4

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).

Abstract

Abstract: We consider a branching random walk in an independent and identically distributed random environment ξ=(ξ_n) indexed by the time. Let W be the limit of the martingale W_n=∫e^-txZ_n(dx)/E_ξ ∫e^-txZ_n(dx), with Z_n denoting the counting measure of particles of generation n, and E_ξ 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

CLC Number:

60K37

Yuejiao WANG, Zaiming LIU, Quansheng LIU, Yingqiu LI. ASYMPTOTIC PROPERTIES OF A BRANCHING RANDOM WALK WITH A RANDOM ENVIRONMENT IN TIME[J].Acta mathematica scientia,Series B, 2019, 39(5): 1345-1362.

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References

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

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