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Pré-Publication, Document De Travail Année : 2022

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

Résumé

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.
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Dates et versions

hal-03251326 , version 1 (26-01-2022)
hal-03251326 , version 2 (17-11-2022)

Identifiants

Citer

Pascal Maillard, Bastien Mallein. On the branching convolution equation $\mathcal E = \mathcal{Z} \circledast \mathcal E$. 2022. ⟨hal-03251326v1⟩
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