Moderate deviation for super-Brownian motion with immigration
type: Item
instance of: scholarly article
title: Moderate deviation for super-Brownian motion with immigration
zbMATH Open document ID: 1081.60024
author: Mei Zhang
published in: Science in China. Series A
publication date: 2005-08-30T00:00:00Z
review text: The author considers \(d\)-dimensional super-Brownian motion with immigration determined by Lebesgue measure \(\lambda\). This is a measure-valued process \(X^\lambda= (X^\lambda_t, \mathbb{P}^\lambda_\mu)\) with transition probabilities given by \[ \mathbb{P}^\lambda_\mu\exp\{-\langle X^\lambda_t, f\rangle\}= \exp\Biggl\{- \langle\mu, V(t,\cdot)\rangle- \int^t_0 \langle\lambda, V(s,\cdot)\rangle\,ds\Biggr\}, \] where \(X^\lambda_0= \mu\), \(f\) belongs to a suitable class of test functions on \(\mathbb{R}^d\), and \(V(\cdot,\cdot)\) is the unique mild solution of \(\partial V(t)/\partial t= \Delta V(t)- V^2(t)\), \(V(0)= f\). A moderate deviation principle is proved for \(X^\lambda_t\) under the normalizations \(t^{1-\delta/4}\) for \(d= 1\), \(t^{1-\delta/2}(\log t)^{\delta/2}\) for \(d= 2\), \(t^{1-\delta/2}\) for \(d\geq 3\), for each \(\delta\in (0,1)\). This principle ``fills the gap'' between the central limit theorem \((\delta= 1)\) and the large deviation principle \((\delta= 0)\) for \(X^\lambda_t\). The speed functions also vary between the corresponding ones for these two results. The proof is based on a series expansion of the cumulant semigroup.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 2200227
zbMATH Keywords: moderate deviation principle
reviewed by: Luis G. Gorostiza
MaRDI profile type: Publication
description: scientific article