Fluctuations of the linear functionals for supercritical non-local branching superprocesses

Suppose {X_t: t ≥ 0} is a supercritical superprocess on a Luzin space E, with a non-local branching mechanism and probabilities P_δx, when initiated from a unit mass at x ∈ E. By “supercritical”, we mean that the first moment semigroup of X_t exhibits a Perron-Frobenius type behaviour characterized by an eigentriplet (λ₁, ϕ, ˜ϕ), where the principal eigenvalue λ₁ is greater than 0. Under a second moment condition, we prove that X_t satisfies a law of large numbers in the sense that, for any bounded measurable function f on E, lim t→+∞^e−λ1t 〈f, X_t〉 = 〈f, ˜ϕ〉W∞^ϕ in L² (P_δx), where W∞^ϕ is the limit of the martingale W^ϕt = e^−λ1t 〈ϕ, X_t〉. The main purpose of this paper is to further investigate the fluctuations of the linear functional e^−λ1t 〈f, X_t〉 around the limit given by the law of large numbers. To this end, we introduce a parameter ɛ(f) for a bounded measurable function f, which determines the exponent term of the decay rate for the first moment of the fluctuation. Qualitatively, the second-order behaviour of 〈f, X_t〉 depends on the sign of ɛ(f) − λ₁/2. We prove that, for a suitable test function f, the fluctuation of the associated linear functional exhibits distinct asymptotic behaviours depending on the magnitude of ɛ(f): If ɛ(f) ≥ λ₁/2, the fluctuation converges in distribution to a Gaussian limit under appropriate normalization; If ɛ(f) < λ₁/2, the fluctuation converges to an L² limit with a larger normalization factor. In particular, when the test function is chosen as the right eigenfunction ϕ, we establish a functional central limit theorem. As an application, we consider a multitype superdiffusion in a bounded domain. For this model, we derive limit theorems for the fluctuations of arbitrary linear functionals.