A stochastic wave equation in two space dimensions: smoothness of the law
type: Item
instance of: scholarly article
title: A stochastic wave equation in two space dimensions: smoothness of the law
zbMATH Open document ID: 0944.60067
author: Annie Millet
author: Marta Sanz-Solé
published in: The Annals of Probability
publication date: 1999-11-09T00:00:00Z
review text: One considers real-valued solutions of stochastic wave equations of the type \[ \left( {\partial^2\over\partial t^2}-\Delta \right) u(t,x) =\sigma(u(t,x))F(dt,dx)+b(u(t,x)) \] with two-dimensional spatial variable \(x\) and given initial conditions \(u(0,x)\) and \({\partial u\over \partial t}u(0,x)\). The noise \(F(dt,dx)\) is assumed to be white in time, and its spatial correlation is described by a function \(f\). Under an integral condition on \(f\), the existence and uniqueness of a solution \(u(t,x)\) is proved. Then the Malliavin calculus is applied to the smoothness of its law.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1359665
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: wave equation
zbMATH Keywords: Gaussian noise
zbMATH Keywords: Malliavin calculus
reviewed by: Jean Picard
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1022677387
OpenAlex ID: W2080675790
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DOI: 10.1214/AOP/1022677387
description: scientific article; zbMATH DE number 1359665