On the branching convolution equation $\mathcal E = \mathcal{Z} \circledast \mathcal E$
Probability
2022-01-04 v1
Abstract
We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by the branching convolution operation introduced by Bertoin and Mallein (2019), and by the law of a random point measure on the real line, we are interested in solutions to the fixed point equation with a random point measure distribution. Under suitable assumptions, we characterize all solutions of this equation as shifted decorated Poisson point processes with a uniquely defined shift.
Keywords
Cite
@article{arxiv.2106.02544,
title = {On the branching convolution equation $\mathcal E = \mathcal{Z} \circledast \mathcal E$},
author = {Pascal Maillard and Bastien Mallein},
journal= {arXiv preprint arXiv:2106.02544},
year = {2022}
}