Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation.
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.
Stochastic partial differential equations
scientific article; zbMATH DE number 2057773