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

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.