A stochastic wave equation in dimension 3: Smoothness of the law
The authors consider the stochastic partial differential equation \[ \left({\partial^2\over\partial t^2}-\Delta_3\right)u(t,x)=\sigma(u(t,x)){\dot F}(t,x)+b(u(t,x)), \qquad u(0,x)={\partial u\over\partial t}(0,x)=0, \] \((t,x)\in(0,T]\times{\mathbb R}^3\), where \(\Delta_3\) is the Laplace operator in \({\mathbb R}^3\), \(\sigma, b\) are Lipschitz functions, \(F\) is a centered Gaussian random field indexed by the test functions from \({\mathcal D}({\mathbb R}^4)\) with the covariance function \(\int_{{\mathbb R}_+}\int_{{\mathbb R}^3}({\mathcal F}\varphi)(s)(\xi)\overline{({\mathcal F}\psi)(s)(\xi)}\mu(d\xi)ds\); \(\mathcal F\) is the Fourier transform and \(\mu\) is a spectral measure. The equation is meant in the mild sense as defined by \textit{R. C. Dalang} and \textit{N. E. Frangos} [Ann. Probab. 26, 187--212 (1998; Zbl 0938.60046)]. The paper is closely related to a paper by the same authors [J. Funct. Anal. 206, 1--32 (2004; Zbl 1039.60054)], where existence of a density of the law of the solution was investigated. In the present paper smoothness of this density is proved (\(u(t,x)\in {\mathbb D}^\infty\)). The Malliavin calculus apparatus is used.
stochastic partial differential equations
scientific article; zbMATH DE number 2072663