Ergodic theory for a superprocess over a stochastic flow
type: Item
instance of: scholarly article
title: Ergodic theory for a superprocess over a stochastic flow
zbMATH Open document ID: 1236.60083
author: Jie Xiong
author: Mei Zhang
author: Zenghu Li
published in: Stochastic Processes and their Applications
publication date: 2010-08-03T00:00:00Z
review text: The paper is connected to the ergodic theory for superprocesses over stochastic flows. A superprocess over a stochastic flow was constructed by \textit{G. Skoulakis} and \textit{R. J. Adler} [Ann. Appl. Probab. 11, No. 2, 488--543 (2001; Zbl 1018.60052)]. Namely, let \(\sigma_1(x)\) and \(\sigma_2(x)\) be \(d\times d\) matrices \((x\in\mathbb R^d)\). Suppose that \(W\) and \(B_1,B_2, \dots\) are independent \(d\)-dimensional Brownian motions. Let us consider a branching particle system on \(\mathbb R^d\) described as follows. Between its branchings, the motion of the \(i\)th particle is defined by the stochastic differential equation \[ {\mathrm d}\xi_i(t) = \sigma_1(\xi_i(t))\,{\mathrm d} W(t) + \sigma_2(\xi_i(t))\,{\mathrm d} B_i(t). \] The particle splits into two or dies with equal probabilities when its standard exponential lifetime runs out, independent of the others. Following the result of Skoulakis and Adler, a suitable scaling limit of the above system gives a continuous superprocess. The authors of the present paper study the long time limiting behaviour of the occupation time of the superprocess over a stochastic flow introduced by Skoulakis and Adler.. The ergodic theorems for dimensions \(d=2\) and \(d\geq 3\) are established. The proofs depend heavily on a characterization of the conditional log-Laplace equation of the occupation time process.
reviewed by: Mátyás Barczy
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J35
zbMATH DE Number: 5763688
zbMATH Keywords: superprocess
zbMATH Keywords: dependent spatial motion
zbMATH Keywords: ergodic theorem
zbMATH Keywords: branching particle system
zbMATH Keywords: non-linear SPDE
zbMATH Keywords: conditional log-Laplace functional
MaRDI profile type: Publication
full work available at URL: j.spa.2010.03.012
OpenAlex ID: W2004252088
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DOI: 10.1016/J.SPA.2010.03.012
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description: scientific article; zbMATH DE number 5763688