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We give a functional limit theorem for the fluctuations of the rescaled occupation time process of a critical branching particle system in R d with symmetric a-stable motion and aodo2a, which leads to a long-range dependence process involving sub-fractional Brownian motion. We also give an analogous result for the system without branching and doa, which involves fractional Brownian motion. We use a space-time random field approach. r 2005 Elsevier B.V. All rights reserved. MSC: primary 60F17; 60J80; secondary 60G20; 60G18

We prove a functional limit theorem for the rescaled occupation time fluctuations of a (d, α, β)-branching particle system [particles moving in R d according to a symmetric α-stable Lévy process, branching law in the domain of attraction of a (1 + β)-stable law, 0 < β < 1, uniform Poisson initial state] in the case of intermediate dimensions, α/β < d < α(1 + β)/β. The limit is a process of the form Kλξ, where K is a constant, λ is the Lebesgue measure on R d , and ξ = (ξt) t≥0 is a (1 + β)-stable process which has long range dependence. For α < 2, there are two long range dependence regimes, one for β > d/(d + α), which coincides with the case of finite variance branching (β = 1), and another one for β ≤ d/(d + α), where the long range dependence depends on the value of β. The long range dependence is characterized by a dependence exponent κ which describes the asymptotic behavior of the codifference of increments of ξ on intervals far apart, and which is d/α for the first case (and for α = 2) and (1 + β − d/(d + α))d/α for the second one. The convergence proofs use techniques of S ′ (R d )-valued processes.

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In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covarianceparameters a > −1, −1 < b ≤ 1, |b| ≤ 1 + a, corresponds to fractional Brownian motion for a = 0, −1 < b < 1. The second one, with covarianceparameter 0 < h ≤ 4, corresponds to sub-fractional Brownian motion for 0 < h < 2. The third one, with covarianceis related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.