Stable Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Abstract

In this paper, we study the asymptotic behavior of a supercritical $(\xi ,\psi )$-superprocess $(X_t{)}_{t\ge 0}$ whose underlying spatial motion $\xi$ is an Ornstein-Uhlenbeck process on ${\mathbb{R}}^d$ with generator $L=\frac{1}{2}{\sigma }^2\Delta -bx\cdot \nabla$ where $\sigma ,b>0$; and whose branching mechanism $\psi$ satisfies Grey's condition and some perturbation condition which guarantees that, when $z\to 0$, $\psi (z)=-\alpha z+\eta z^{1+\beta }(1+o(1))$ with $\alpha >0$, $\eta >0$ and $\beta \in (0,1)$. Some law of large numbers and $(1+\beta )$-stable central limit theorems are established for $(X_t(f){)}_{t\ge 0}$, 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.