Issue Information
روزنامه با شماره پیاپی سال 2004
Pages
32
From page
1
To page
32
Abstract
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.
Journal title
Journal of Functional Analysis
Serial Year
2004
Journal title
Journal of Functional Analysis
Record number
709239
Link To Document