On the necessary and sufficient conditions to solve a heat equation with general additive Gaussian noise

TL;DR

This paper establishes necessary and sufficient conditions on additive Gaussian noise for solving the stochastic heat equation, using variance and Besov space methods, and explores solution regularity.

Contribution

It provides a unified characterization of noise conditions for solving the stochastic heat equation and compares two different analytical approaches.

Findings

Both methods yield the same noise conditions.

Path-wise approach reveals solution regularity.

Conditions are applicable to general additive Gaussian noise.

Abstract

In this note we consider stochastic heat equation with general additive Gaussian noise. Our aim is to derive some necessary and sufficient conditions on the Gaussian noise in order to solve the corresponding heat equation. We investigate this problem invoking two different methods, respectively based on variance computations and on path-wise considerations in Besov spaces. We are going to see that, as anticipated, both approaches lead to the same necessary and sufficient condition on the noise. In addition, the path-wise approach brings out regularity results for the solution.