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Suppose that X={X t :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+βλ 2+∫(0,+∞)(e −λx −1+λx)n(dx), where α=−ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫(0,+∞) x 2 n(dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W t =e −αt ∥X t ∥ is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W ∞ such that W t →W ∞, $\mathbb{P} _{\mu}$ -a.s. as t→∞. In this paper we establish some spatial central limit theorems for X.
Acta Applicandae Mathematicae – Springer Journals
Published: Aug 8, 2013
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