Study of a SPDE driven by a Poisson noise
type: Item
instance of: scholarly article
title: Study of a SPDE driven by a Poisson noise
zbMATH Open document ID: 0939.60064
author: Erwan Saint Loubert Bié
publication date: 2000-07-05T00:00:00Z
review text: Let \(U\) be a normed space. The author considers a stochastic parabolic differential equation on \(\mathbb{R}^+\times\mathbb{R}^d\times U\) driven by some time-space Poissonian noise \(\lambda^+\) \[ \begin{aligned} V(t,x)= &\;u_0(x)+\int^t_0 {1 \over 2}\Delta V(s,x)ds +\int^t_0 g\bigl(V(s,x),s,x\bigr)ds\\ & +\int^t_0 \int_{h \in U} f\bigl(V_-(s,x),s,x,h)\bigr) d\lambda^+(s,x,h).\end{aligned} \] He uses the contraction principle in order to show that this initial value problem has a unique solution if the intensity of \(\lambda^+\) is suitably bounded and if the functions \(f:\mathbb{R}\times \mathbb{R}^+\times \mathbb{R}^d\times U\) and \(g:\mathbb{R} \times\mathbb{R}^+ \times\mathbb{R}^d\) are Lipschitzian and suitably bounded. He also gives practicable conditions so that this solution is \(l\)-times continuously differentiable in space or time.
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 1182322
zbMATH Keywords: stochastic parabolic differential equation
zbMATH Keywords: Poissonian noise
zbMATH Keywords: existence and uniqueness of solutions
zbMATH Keywords: regularity of solutions
reviewed by: Gunter Ritter
MaRDI profile type: Publication
full work available at URL: s004400050169
OpenAlex ID: W2039279472
DOI: 10.1007/S004400050169
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1182322