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

摘要

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 mu satisfies that integralR(3)mu(dxi)(1+xi(2))(-eta) < infinity, for some eta is an element of (0, 1/2). Our approach is based on the mild formulation of the equation given by means of Dalang's extended version of Walsh's stochastic integration; we use the tools of Malliavin calculus. Let S-3 be the fundamental solution to the 3-dimensional wave equation. The assumption on the noise yields upper and lower bounds for the integral integral(0)(t) ds integral(R3) mu(dxi)FS3(s)(xi)(2) and upper bounds for integral(0)(t) ds integral(R3mu)(dxi)xiFS3(s)(xi)(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. (C) 2003 Elsevier Inc. All rights reserved. [References: 23]