Stable central limit theorems for super Ornstein-Uhlenbeck processes

摘要

In this paper, we study the asymptotic behavior of a supercritical $(xi ,psi )$-superprocess $(X_{t})_{tgeq 0}$ whose underlying spatial motion $xi $ is an Ornstein-Uhlenbeck process on $mathbb{R} ^{d}$ with generator $L = rac{1} {2}sigma ^{2}Delta - b x cdot abla $ where $sigma , b 0$; and whose branching mechanism $psi $ satisfies Grey’s condition and a perturbation condition which guarantees that, when $zo 0$, $psi (z)=-lpha z + eta z^{1+eta } (1+o(1))$ with $lpha 0$, $eta 0$ and $eta in (0, 1)$. Some law of large numbers and $(1+eta )$-stable central limit theorems are established for $(X_{t}(f) )_{tgeq 0}$, 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 to the branching rate being relatively small, large or critical at a balanced value.