SPDEs with time-independent L\'evy colored noise

TL;DR

This paper introduces a time-independent Le9vy colored noise for SPDEs, analyzing solution existence, moment conditions, and applying Malliavin calculus to heat and wave equations across dimensions.

Contribution

It develops a novel time-independent Le9vy colored noise model and studies solution existence and moment properties for related SPDEs.

Findings

Established conditions for finite p-th moments of solutions.

Applied Malliavin calculus to SPDEs with multiplicative noise.

Analyzed heat and wave equations in all dimensions.

Abstract

In this article, we introduce a time-independent version of the L\'evy colored noise considered in Balan (2015) and Balan and Jim\'enez (2026). We study the existence of the solution of a linear stochastic partial differential equation with this type of noise, and we identify some necessary conditions which guarantee that the solution has finite $p$-th order moments. Using tools from Malliavin calculus, we investigate the existence of the solution for the equation with multiplicative noise. As examples, we consider the stochastic heat and wave equations in any dimension $d \geq 1$.