Integration with respect to Levy colored noise, with applications to SPDEs

In this article, we introduce a Levy analogue of the spatially homogeneous Gaussian noise of [5], and we construct a stochastic integral with respect to this noise. The spatial covariance of the noise is given by a tempered measure mu on R-d, whose density is given by vertical bar h vertical bar(2) for a symmetric complex-valued function h. Without assuming that the Fourier transform of m is a non-negative function, we identify a large class of integrands with respect to this noise. As an application, we examine the linear stochastic heat and wave equations driven by this type of noise.