An It\^o type formula for the additive stochastic heat equation

TL;DR

This paper develops an Itô formula for the one-dimensional stochastic heat equation driven by space-time white noise, using regularity structures to connect stochastic calculus with distributional solutions.

Contribution

It introduces a novel Itô formula for stochastic PDE solutions via regularity structures, bridging stochastic calculus and distribution theory.

Findings

Derived an Itô formula for the stochastic heat equation

Expressed solutions in terms of reconstructed modelled distributions

Connected regularity structures with classical stochastic calculus

Abstract

We use the theory of regularity structures to develop an It\^o formula for $u$, the solution of the one dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular for any smooth enough function $\varphi$ we can express the random distribution $(\partial_t-\partial_{xx})\varphi(u)$ and the random field $\varphi(u)$ in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of stochastic calculus.