Absolute continuity of the law of the solution to the 3-dimensional stochastic wave equation

We present new results regarding the existence of density of the real-valued solution to a 3-dimensional stochastic wave equation. The noise is white in time and with a spatially homogeneous correlation whose spectral measure μ satisfies that ∫R3μ(dξ)(1+|ξ|2)−η<∞, for some η∈(0,12). Our approach is based on the mild formulation of the equation given by means of Dalangʹs extended version of Walshʹs stochastic integration; we use the tools of Malliavin calculus. Let S3 be the fundamental solution to the 3-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral ∫0tds∫R3μ(dξ)|FS3(s)(ξ)|2 and upper bounds for ∫0tds∫R3μ(dξ)|ξ||FS3(s)(ξ)|2 in terms of powers of t. These estimates are crucial in the analysis of the Malliavin variance, which can be done by a comparison procedure with respect to smooth approximations of the distribution-valued function S3(t) obtained by convolution with an approximation of the identity.