Acta Applicandae Mathematicae

, Volume 130, Issue 1, pp 9–49

# Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Article

DOI: 10.1007/s10440-013-9837-0

Ren, YX., Song, R. & Zhang, R. Acta Appl Math (2014) 130: 9. doi:10.1007/s10440-013-9837-0

## Abstract

Suppose that X={Xt:t≥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 ψ(λ)=−αλ+βλ2+∫(0,+∞)(eλx−1+λx)n(dx), where α=−ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫(0,+∞)x2n(dx)<+∞. Let $$\mathbb{P} _{\mu}$$ be the law of X with initial measure μ. Then the process Wt=eαtXt∥ is a positive $$\mathbb{P} _{\mu}$$-martingale. Therefore there is W such that WtW, $$\mathbb{P} _{\mu}$$-a.s. as t→∞. 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 γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(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 has the same distribution as W conditioned on no-extinction and $$G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$$. Moreover, W and G1(f) are independent. In the critical rate case α=2γ(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 has the same distribution as W conditioned on no-extinction and $$G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$$. Moreover W and G2(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(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.

### Keywords

Central limit theorem Backbone decomposition Superprocess Super Ornstein-Uhlenbeck process Branching process Branching Ornstein-Uhlenbeck process Ornstein-Uhlenbeck process

### Mathematics Subject Classification (2000)

60J80 60G57 60J45