Occupation Time Fluctuations in Branching Systems

2000/05/23 by Don Dawson, Dawson, Don, Luis G. Gorostiza +5
Computer Science · Mathematics · Physics and Astronomy · #Cellular Automata and Applications #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR

paper · pdf · doi:10.48550/arxiv.math/0005231

59 pages

arxiv created 2000/05/23 · openalex publication_date 2000/05/23 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider particle systems in locally compact Abelian groups with particles moving according to a process with symmetric stationary independent increments and undergoing one and two levels of critical branching. We obtain long time fluctuation limits for the occupation time process of the one-and two-level systems. We give complete results for the case of finite variance branching, where the fluctuation limits are Gaussian random fields, and partial results for an example of infinite variance branching, where the fluctuation limits are stable random fields. The asymptotics of the occupation time fluctuations are determined by the Green potential operator G of the individual particle motion and its powers G², G³, and by the growth as t→∞ of the operator G_t=∫^t₀T_sds and its powers, where T_t is the semigroup of the motion. The results are illustrated with two examples of motions: the symmetric α-stable Lévy process in \erre^d (0

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