Linear SPDEs with harmonizable noise

TL;DR

This paper introduces a new class of harmonizable noise processes for linear SPDEs, extending existing models to include non-Gaussian and fractional noises, and establishes conditions for solutions to heat and wave equations.

Contribution

It develops a framework for analyzing linear SPDEs driven by harmonizable noise, broadening the scope beyond Gaussian cases and including fractional noises with H<1/2.

Findings

Established existence conditions for solutions of SPDEs with harmonizable noise.

Extended results to fractional powers of the Laplacian, including H<1/2.

Unified treatment of Gaussian and non-Gaussian harmonizable noises.

Abstract

Using tools from the theory of random fields with stationary increments, we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise (called harmonizable) is not necessarily Gaussian, but it includes the spatially homogeneous Gaussian noise introduced in Dalang (1999), and the fractional noise considered in Balan and Tudor (2010). We derive some general conditions for the existence of a random field solution of a linear SPDE with harmonizable noise, under some mild conditions imposed on the Green function of the differential operator which appears in this equation. This methodology is applied to the study of the heat and wave equations (possibly replacing the Laplacian by one of its fractional powers), extending in this manner the results of Balan and Tudor (2010) to the case $H<1/2$.