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Invariant measures of critical branching random walks in high dimension. (English) Zbl 1509.60155

Summary: In this work, we characterize cluster-invariant point processes for critical branching spatial processes on \({\mathbb{R}^d}\) for all large enough \(d\) when the motion law is \(\alpha \)-stable or has a finite discrete range. More precisely, when the motion is \(\alpha \)-stable with \(\alpha \le 2\) and the offspring law \(\mu\) of the branching process has an heavy tail such that \(\mu (k)\sim{k^{-2-\beta }} \), then we need the dimension \(d\) to be strictly larger than the critical dimension \(\alpha / \beta \). In particular, when the motion is Brownian and the offspring law \(\mu\) has a second moment, this critical dimension is 2. Contrary to the previous work of M. Bramson et al. in [Ann. Probab. 25, No. 1, 56–70 (1997; Zbl 0882.60091)] whose proof used PDE techniques, our proof uses probabilistic tools only.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)

Citations:

Zbl 0882.60091
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References:

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