Covariance structure of parabolic stochastic partial differential equations with multiplicative L\'evy noise

TL;DR

This paper derives a deterministic equation for the covariance function of solutions to parabolic SPDEs with multiplicative Lévy noise, enabling better understanding of their second moments through a variational approach.

Contribution

It introduces a novel variational framework for the covariance function of parabolic SPDEs with Lévy noise, linking stochastic properties to deterministic equations.

Findings

Derived a well-posed deterministic variational problem for the covariance function.

Established a deterministic equation for the second moment of the solution.

Provided a mathematical foundation for analyzing covariance structures in Lévy-driven SPDEs.

Abstract

The characterization of the covariance function of the solution process to a stochastic partial differential equation is considered in the parabolic case with multiplicative L\'evy noise of affine type. For the second moment of the mild solution, a well-posed deterministic space-time variational problem posed on projective and injective tensor product spaces is derived, which subsequently leads to a deterministic equation for the covariance function.