On a new version of the Ito's formula for the stochastic heat equation

TL;DR

This paper develops a new version of Itô's formula specifically for the one-dimensional stochastic heat equation driven by space-time white noise, using elementary transform properties and explicit solution representations.

Contribution

It introduces a novel Itô's-type formula for the stochastic heat equation, expanding the theoretical framework for stochastic partial differential equations.

Findings

Derived an explicit Itô's-type formula for the stochastic heat equation

Established connections with existing Itô formulas in literature

Provided a proof based on the $\\mathcal{S}$-transform and solution representation

Abstract

We derive an It\^o's-type formula for the one dimensional stochastic heat equation driven by a space-time white noise. The proof is based on elementary properties of the $\mathcal{S}$-transform and on the explicit representation of the solution process. We also discuss the relationship with other versions of this It\^o's-type formula existing in literature.