In this work, we characterize cluster-invariant point processes for critical branching spatial processes on 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 and the offspring law μ of the branching process has an heavy tail such that , then we need the dimension d to be strictly larger than the critical dimension . 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  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.
"Invariant measures of critical branching random walks in high dimension." Electron. J. Probab. 28 1 - 38, 2023. https://doi.org/10.1214/23-EJP906