2019/03/09 by Ren, Yan-Xia, Song, Renming, Sun, Zhenyao +1
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1903.03751
In this paper, we study the asymptotic behavior of a supercritical (ξ,ψ)-superprocess (Xt)t≥ 0 whose underlying spatial motion ξ is an Ornstein-Uhlenbeck process on \mathbb Rd with generator L = (1)/(2)σ2Δ- b x ⋅ ∇ where σ, b >0; and whose branching mechanism ψ satisfies Grey's condition and some perturbation condition which guarantees that, when z→ 0, ψ(z)=-αz + ηz1+β (1+o(1)) with α> 0, η>0 and β∈ (0, 1). Some law of large numbers and (1+β)-stable central limit theorems are established for (Xt(f) )t≥ 0, where the function f is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding the branching rate being relatively small, large or critical at a balanced value.