Yaglom theorem for critical branching random walk on ℤ^d

2025/12/30 by Xinxin Chen, Shen Lin
Mathematics · #math.PR

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Abstract

We study the critical branching random walk on ℤ^d started from a distant point x and conditioned to hit some compact set K in ℤ^d. We are interested in the occupation time in K and present its asymptotic behaviors in different dimensions. It is shown in this work that the occupation time is of order ‖x‖^4-d in dimensions d≤ 3, of order log‖x‖ in dimension d=4, and of order 1 in dimensions d≥ 5. The corresponding weak convergences are also established. These results answer a question raised by Le Gall and Merle (Elect. Comm. in Probab. 11 (2006), 252-265).

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