On the branching convolution equation E=Z⊛E

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, 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 $\mathcal{E}=\mathcal{Z}⊛\mathcal{E}$, 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.