Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation.
type: Item
instance of: scholarly article
title: Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation.
zbMATH Open document ID: 1039.60054
DOI: 10.1016/S0022-1236(03)00065-X
author: Lluís Quer-Sardanyons
author: Marta Sanz-Solé
published in: Journal of Functional Analysis
publication date: 2004-03-15T00:00:00Z
review text: The authors study the existence of density of the real-valued solution to a 3-dimensional wave equation, \[ \Biggl({\partial^2 \over\partial t^2}- \Delta_3\Biggr) u(t,x)= \sigma(u(t, x))\dot F(t, x) + b(u(t, x)),\quad u(0, x)= {\partial u\over\partial t}\, (0, x)= 0, \] where \((t, x)\in (0,T]\times \mathbb{R}^3\), \(T> 0\), and \(\Delta_3\) denotes the Laplacian operator on \(\mathbb{R}^3\). The noise is white in time and with a spatially homogeneous correlation whose spectral measure \(\mu\) satisfies \(\int_{\mathbb{R}^3} \mu(d\xi)(1+ |\xi|)^2)^{-\eta}< \infty\), for some \(\eta\in (0,{1\over 2})\). The rigorous definition in the mild form of the solution is given using Dalang's extended version of Walsh's stochastic integration. The basic tool in order to prove the existence of density is the Malliavin calculus. To give a rigorous meaning to the equation satisfied by Malliavin derivative of \(u(t, x)\), the authors extend Dalang's stochastic integral to integrators that are defined by stochastic integration of Hilbert-valued predictable processes. On the other hand, the authors can solve the problems caused by the fact that \(S_3(t)\), the fundamental solution to the 3-dimensional wave equation, is a distribution using smooth approximations obtained by convolution with an approximation of the density.
reviewed by: Carles Rovira
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2057773
zbMATH Keywords: Malliavin calculus
zbMATH Keywords: Stochastic partial differential equations
zbMATH Keywords: Wave equation
MaRDI profile type: Publication
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full work available at URL: s0022-1236(03)00065-x
OpenAlex ID: W1976559645
description: scientific article; zbMATH DE number 2057773