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Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

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References (28)

Publisher
Springer Journals
Copyright
Copyright © 2013 by Springer Science+Business Media Dordrecht
Subject
Mathematics; Mathematics, general; Computer Science, general; Theoretical, Mathematical and Computational Physics; Statistical Physics, Dynamical Systems and Complexity; Mechanics
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-013-9837-0
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Aug 8, 2013

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