Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={Xt:t驴0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism 驴(驴)=驴驴驴+脽驴2+驴(0,+驴)(e驴驴x驴1+驴x)n(dx), where 驴=驴驴驴(0+)0, 脽驴0, and n is a measure on (0,驴) such that 驴(0,+驴)x2n(dx)$\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process Wt=e驴驴t驴Xt驴 is a positive $\mathbb{P} _{\mu}$-martingale. Therefore there is W驴 such that Wt驴W驴, $\mathbb{P} _{\mu}$-a.s. as t驴驴. In this paper we establish some spatial central limit theorems for X.Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer 驴(f) in term of the spectral decomposition of f. In the small branching rate case 驴驴(f)b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W驴 has the same distribution as W驴 conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$. Moreover, W驴 and G1(f) are independent. In the critical rate case 驴=2驴(f)b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W驴 has the same distribution as W驴 conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$. Moreover W驴 and G2(f) are independent. We also establish two central limit theorems in the large branching rate case 驴2驴(f)b.Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Miłoś in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Miłoś. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya's 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960's and early 1970's.