Covariance structure of parabolic stochastic partial differential equations

TL;DR

This paper derives a deterministic tensorized evolution equation for the second moment and covariance of solutions to parabolic stochastic PDEs driven by Wiener processes, establishing well-posedness of the formulation.

Contribution

It introduces a novel tensorized evolution equation for the second moment and covariance of parabolic stochastic PDE solutions, along with a proof of well-posedness.

Findings

Derived a deterministic tensorized evolution equation for covariance.

Proved well-posedness of the space-time weak variational formulation.

Abstract

In this paper parabolic random partial differential equations and parabolic stochastic partial differential equations driven by a Wiener process are considered. A deterministic, tensorized evolution equation for the second moment and the covariance of the solutions of the parabolic stochastic partial differential equations is derived. Well-posedness of a space-time weak variational formulation of this tensorized equation is established.