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The Spread Speed of Multiple Catalytic Branching Random Walks

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Abstract

In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk (Xn) on ℤ. If Mn is its maximal position at time n, we prove that there is a constant α > 0 such that Mn/n converges to α almost surely on the set of infinite number of visits to the set of catalysts. We also derive the asymptotic law of the centered process Mn — αn as n → ∞. Our results are similar to those in [13]. However, our results are proved under the assumption of finite L log L moment instead of finite second moment. We also study the limit of (Xn) as a measure-valued Markov process. For any function f with compact support, we prove a strong law of large numbers for the process Xn(f).

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Acknowledgment

The author would like to thank Professor Yanxia Ren and Professor Renming Song for their valuable comments motivating this work. The author wish to thank two anonymous referees for their careful readings and helpful comments on the first version of this paper.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

    Rong-li Liu

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  1. Rong-li Liu
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Correspondence to Rong-li Liu.

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This paper is supported in part by the National Natural Science Foundation of China (No.12271374).

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Liu, Rl. The Spread Speed of Multiple Catalytic Branching Random Walks. Acta Math. Appl. Sin. Engl. Ser. 39, 262–292 (2023). https://doi.org/10.1007/s10255-023-1046-7

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