A stochastic wave equation in dimension 3: smoothness of the law

Abstract

We prove the existence and regularity of the density of the real-valued solution to a three-dimensional stochastic wave equation. The noise is white in time and has a spatially homogeneous correlation whose spectral measure μ satisfies , for some $\eta \in (0,\frac{1}{2})$. Our approach uses the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration. We apply the tools of Malliavin calculus on the appropriate Gaussian space related to the noise. An extension of Dalang's stochastic integral to the Hilbert-valued setting is needed. Let S₃ be the fundamental solution to the three-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral ${\int }_0^{t}ds{\int }_{R^3}\mu (d\xi )\left|calF{S}_3\right(s\left)\right(\xi ){|}^2$ and upper bounds for ${\int }_0^{t}ds{\int }_{R^3}\mu (d\xi )\left|\xi \right|\left|calF{S}_3\right(s\left)\right(\xi ){|}^2$ in terms of powers of t. These estimates, together with a suitable mollifying procedure for S₃, are crucial in the analysis of the inverse of the Malliavin variance.