Occupation time fluctuations of an infinite-variance branching system in large dimensions

Abstract

We prove limit theorems for rescaled occupation time fluctuations of a (d,alpha,beta)-branching particle system (particles moving in R^d according to a spherically symmetric alpha-stable Levy process, (1+beta)-branching, 0<beta<1, uniform Poisson initial state), in the cases of critical dimension, d=alpha(1+beta)/beta, and large dimensions, d>alpha(1+beta)/beta. The fluctuation processes are continuous but their limits are stable processes with independent increments, which have jumps. The convergence is in the sense of finite-dimensional distributions, and also of space-time random fields (tightness does not hold in the usual Skorohod topology). The results are in sharp contrast with those for intermediate dimensions, alpha/beta < d < d(1+beta)/beta, where the limit process is continuous and has long range dependence (this case is studied by Bojdecki et al, 2005). The limit process is measure-valued for the critical dimension, and S'(R^d)-valued for large dimensions. We also raise some questions of interpretation of the different types of dimension-dependent results obtained in the present and previous papers in terms of properties of the particle system.