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

Abstract

In this article, we introduce a Lévy 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 μ on , whose density is given by for a symmetric complex-valued function h. Without assuming that the Fourier transform of μ 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.

Keywords::

• Lévy processes
• stochastic integral
• stochastic heat equation
• stochastic wave equation

MSC (2010) Classification::

• Primary 60G51
• secondary 60H15

Acknowledgements

The author is grateful to Robert Dalang for suggesting this problem, and would also like to thank an anonymous referee for carefully reading the paper.

Additional information

Funding

This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada under [grant no. 263899-2012].