Occupation Time Fluctuations in Branching Systems

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 \(G_t = \int_0^t {T_s } ds\)and its powers, where Tt is the semigroup of the motion. The results are illustrated with two examples of motions: the symmetric α-stable Levy process in \(\mathbb{R}^d (0 < \alpha \leqslant 2)\), and the so called c-hierarchical random walk in the hierarchical group of order N (0<c<N). We show that the two motions have analogous asymptotics of Gt and its powers that depend on an order parameter γ for their transience/recurrence behavior. This parameter is γ=d/α−1 for the α-stable motion, and γ=log c/log(N/c) for the c-hierarchical random walk. As a consequence of these analogies, the asymptotics of the occupation time fluctuations of the corresponding branching particle systems are also analogous. In the case of the c-hierarchical random walk, however, the growth of Gt and its powers is modulated by oscillations on a logarithmic time scale.