2025/03/23 by Yang, Ting · 1 citation
#60F05 #60G57 #60J68 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2503.17929
Suppose \Xt: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 Xt exhibits a Perron-Frobenius type behaviour characterized by an eigentriplet (λ1,φ,\widetildeφ), where the principal eigenvalue λ1 is greater than 0. Under a second moment condition, we prove that Xt satisfies a law of large numbers. The main purpose of this paper is to further investigate the fluctuations of the linear functional e^-λ1t⟨ f,Xt⟩ around the limit given by the law of large numbers. To this end, we introduce a parameter ε(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,Xt⟩ depends on the sign of ε(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 ε(f): If ε(f)≥ λ1/2, the fluctuation converges in distribution to a Gaussian limit under appropriate normalization; If ε(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.