Open Access
2023 Invariant measures of critical branching random walks in high dimension
Valentin Rapenne
Author Affiliations +
Electron. J. Probab. 28: 1-38 (2023). DOI: 10.1214/23-EJP906


In this work, we characterize cluster-invariant point processes for critical branching spatial processes on Rd for all large enough d when the motion law is α-stable or has a finite discrete range. More precisely, when the motion is α-stable with α2 and the offspring law μ of the branching process has an heavy tail such that μ(k)k2β, then we need the dimension d to be strictly larger than the critical dimension αβ. In particular, when the motion is Brownian and the offspring law μ has a second moment, this critical dimension is 2. Contrary to the previous work of Bramson, Cox and Greven in [4] whose proof used PDE techniques, our proof uses probabilistic tools only.


I would like to thank my Ph.D supervisor, Xinxin Chen, for her very useful pieces of advice without which this work could not have been carried out. I also want to thank Christophe Garban and Hui He whose fruitful discussions with Xinxin Chen have enlightened us a lot.


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Valentin Rapenne. "Invariant measures of critical branching random walks in high dimension." Electron. J. Probab. 28 1 - 38, 2023.


Received: 7 July 2022; Accepted: 16 January 2023; Published: 2023
First available in Project Euclid: 23 January 2023

MathSciNet: MR4538350
zbMATH: 1509.60155
Digital Object Identifier: 10.1214/23-EJP906

Primary: 60

Keywords: branching random walks , Invariant measures , Point processes

Vol.28 • 2023
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