On the branching convolution equation $\mathcal E = \mathcal{Z} \circledast \mathcal E$

TL;DR

This paper characterizes all solutions to a fixed point equation involving a branching convolution operation, identifying them as shifted decorated Poisson point processes under certain conditions.

Contribution

It provides a complete characterization of all solutions to the branching convolution fixed point equation, extending understanding of stable point measures under branching operations.

Findings

Solutions are shifted decorated Poisson point processes.

All solutions are characterized under certain assumptions.

The shift in the processes is uniquely determined.

Abstract

We characterize all random point measures which are in a certain sense stable under the action of branching. Denoting by $\circledast$ the branching convolution operation introduced by Bertoin and Mallein (2019), and by $\mathcal{Z}$ the law of a random point measure on the real line, we are interested in solutions to the fixed point equation \[ \mathcal E = \mathcal{Z} \circledast \mathcal E, \] with $\mathcal E$ 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.