Measure-valued diffusions and stochastic equations with Poisson process
type: Item
instance of: scholarly article
title: Measure-valued diffusions and stochastic equations with Poisson process
zbMATH Open document ID: 1065.60108
author: Zongfei Fu
author: Zenghu Li
published in: Osaka Journal of Mathematics
publication date: 2005-02-18T00:00:00Z
review text: Let \(\{X(t)\}\) be an \((A,b,c)\)-superprocess, \(N(ds; dx, du,dw)\) a Poisson random measure on \([0,\infty)\times E\times [0,\infty)\times W\) with intensity \(ds\,m(dx) duQ^x(dw)\), where \(E\) is a locally compact metric space, \(W\) the set of Wiener excursion paths starting at \(0\). Assuming that \(X\) and \(N\) are independent, it is proved that the equation \[ Y(t)= X(t)+ \int_{[0,t]} \int_E \int_{[0,q(Y(s), x)]} \int_W w(t-s)\,N(ds; dx, du, dw),\qquad t\geq 0, \] has a pathwise unique solution, the distribution of which solves the martingale problem given by \[ \begin{gathered} M_t(f)= w_t(f)- \mu(f)- \int_{[0,t]} w_s(Af- bf)\,ds- \int_{[0,t]} m(q(w_s,\cdot)\,ds,\\ \langle M(f)\rangle_t= \int_{[0,t]} w_s(cf^2)\,ds,\qquad t\geq 0.\end{gathered} \] Here \(q\) is a nonnegative Borel function on \(M(E)\times E\), \(M(E)\) the space of finite Borel measures equipped with the topology of weak convergence, and \(q\) can be interpreted as an interactive immigration rate relative to \(m\).
reviewed by: Heinrich Hering
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 2134876
zbMATH Keywords: superprocess
zbMATH Keywords: interactive immigration
zbMATH Keywords: measure-valued excursions
zbMATH Keywords: Poisson random measure
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2134876