A long range dependence stable process and an infinite variance branching system

Abstract

We prove a functional limit theorem for the rescaled occupation time fluctuations of a $(d,\alpha,\beta)$-branching particle system [particles moving in $\mathbb {R}^d$ according to a symmetric $\alpha$-stable L\'{e}vy process, branching law in the domain of attraction of a $(1+\beta)$-stable law, $0<\beta<1$, uniform Poisson initial state] in the case of intermediate dimensions, $\alpha/\beta<d<\alpha(1+\beta)/\beta$. The limit is a process of the form $K\lambda\xi$, where $K$ is a constant, $\lambda$ is the Lebesgue measure on $\mathbb {R}^d$, and $\xi=(\xi_t)_{t\geq0}$ is a $(1+\beta)$-stable process which has long range dependence. For $\alpha<2$, there are two long range dependence regimes, one for $\beta>d/(d+\alpha)$, which coincides with the case of finite variance branching $(\beta=1)$, and another one for $\beta\leq d/(d+\alpha)$, where the long range dependence depends on the value of $\beta$. The long range dependence is characterized by a dependence exponent $\kappa$ which describes the asymptotic behavior of the codifference of increments of $\xi$ on intervals far apart, and which is $d/\alpha$ for the first case (and for $\alpha=2$) and $(1+\beta-d/(d+\alpha))d/\alpha$ for the second one. The convergence proofs use techniques of $\mathcal{S}'(\mathbb {R}^d)$-valued processes.