A stochastic fixed point equation related to weighted branching with deterministic weights.(English)Zbl 1110.60080

Summary: For real numbers $$C,T_{1},T_{2},...$$ we find all solutions $$\mu$$ to the stochastic fixed point equation $$W\overset \text{d} =\sum_{j\geq 1}T_{j}W_{j}+C$$, where $$W,W_{1},W_{2},\dots$$ are independent real-valued random variables with distribution $$\mu$$ and $$\overset \text{d} =$$ means equality in distribution. All solutions are infinitely divisible. The set of solutions depends on the closed multiplicative subgroup of $${\mathbb R}_{*}={\mathbb R}\backslash\{0\}$$ generated by the $$T_{j}$$. If this group is continuous, i.e. $${\mathbb R}_{*}$$ itself or the positive halfline $${\mathbb R}_{+}$$, then all nontrivial fixed points are stable laws. In the remaining (discrete) cases further periodic solutions arise. A key observation is that the Lévy measure of any fixed point is harmonic with respect to $$\Lambda=\sum_{j\geq 1}\delta_{T_{j}}$$, i.e. $$\Gamma=\Gamma\star\Lambda$$, where $$\star$$ means multiplicative convolution. This will enable us to apply the powerful Choquet-Deny theorem.

MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60E07 Infinitely divisible distributions; stable distributions 60E10 Characteristic functions; other transforms
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