Integration with respect to L\'evy colored noise, with applications to SPDEs

TL;DR

This paper introduces a Lévy-based spatially homogeneous noise, constructs a stochastic integral for it, and applies it to analyze linear stochastic heat and wave equations, broadening the scope of noise models in SPDEs.

Contribution

It develops a Lévy analogue of Gaussian noise, constructs the associated stochastic integral without non-negativity assumptions, and applies it to SPDEs.

Findings

Constructed a stochastic integral with respect to Lévy colored noise.

Identified a broad class of integrands for this noise.

Applied the framework to linear stochastic heat and wave equations.

Abstract

In this article, we introduce a L\'evy analogue of the spatially homogeneous Gaussian noise of Dalang (1999), 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 $\bR^d$, whose density is given by $|h|^2$ for a complex-valued function $h$. Without assuming that the Fourier transform of $\mu$ 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.