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**Particle systems with quasi-homogeneous initial states and their occupation time fluctuations.**
*(English)*
Zbl 1226.60048

Summary: We consider particle systems in \(\mathbb R\) with initial configurations belonging to a class of measures that obey a quasi-homogeneity property, which includes as special cases homogeneous Poisson measures and many deterministic measures (simple example: one atom at each point of \(\mathbb Z\)). The particles move independently according to an alpha-stable Lévy process, \(\alpha>1\), and we also consider the model where they undergo critical branching. Occupation time fluctuation limits of such systems have been studied in the Poisson case. For a branching system in “low” dimension, the limit was characterized by a process called sub-fractional Brownian motion, and this process was attributed to the branching because it had only appeared in that case. In the present more general framework, the sub-fractional Brownian motion is more prevalent, namely, it also appears as a component of the limit for a system without branching in “low” dimension. A new method of proof, based on the central limit theorem, is used.

### MSC:

60F17 | Functional limit theorems; invariance principles |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60G18 | Self-similar stochastic processes |

60G52 | Stable stochastic processes |