Ergodic theorems for linear growth processes with diffusion
type: Item
instance of: scholarly article
title: Ergodic theorems for linear growth processes with diffusion
zbMATH Open document ID: 0682.60093
author: Xiaogu Zheng
author: Wan-Ding Ding
published in: Chinese Annals of Mathematics. Series B
publication date: 1989-01-01T00:00:00Z
review text: Let \({\mathbb{E}}\) be the set of all \(\eta\) : \({\mathbb{Z}}^ d\to {\mathbb{Z}}_+\). The pregenerator of the process studied in the paper is as follows: \[ \Omega f(\eta)=\sum_{x\in {\mathbb{Z}}^ d}\{(\lambda_ 1\eta (x)+\lambda_ 3)[f[\eta +e_ x)-f(x)]+\lambda_ 2\eta (x)[f(\eta -e_ x)-f(x)]\} \] \[ +\sum_{x,y\in {\mathbb{Z}}^ d}\eta (x)p(x,y)[f(\eta -e_ x+e_ y)-f(\eta)] \] where \(\lambda_ 1,\lambda_ 2,\lambda_ 3\geq 0\), \(e_ x\in {\mathbb{E}}\) is the configuration valued one at x and zero elsewhere, (p(x,y)) is a transition probability matrix on \({\mathbb{Z}}^ d\). Suppose that \[ \sup_{x}\sum_{y}p(y,x)=K<\infty. \] One of the main results in the paper for the case \(\lambda_ 3=0\) claims that the process is ergodic if and only if \(\lambda_ 1-\lambda_ 2+K-1<0\) and it is not if \(\lambda_ 1-\lambda_ 2>0\) plus \(\inf_{t>0}\sum_{u}p(t,u,x)>0\) for some x, where \[ p(t,x,u)=\sum^{\infty}_{n=0}(t^ n/n!)p^{(n)}(u,x). \] Moreover, the authors study the construction of the invariant measures in the particular case that \(\lambda_ 3=0\) and \(\lambda_ 1=\lambda_ 2>0\). For the case that \(\lambda_ 3>0\), the ergodic and non-ergodic regions are also described.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4117623
zbMATH Keywords: linear growth processes
zbMATH Keywords: invariant measures
zbMATH Keywords: ergodic and non-ergodic regions
reviewed by: Mu-Fa Chen
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4117623