An Itô type formula for the additive stochastic heat equation

Abstract

We use the theory of regularity structures to develop an Itô 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 $\phi$ we can express the random distribution $({\partial }_t-{\partial }_{xx})\phi (u)$ and the random field $\phi (u)$ in terms of the reconstruction of some modelled distributions. The resulting objects are then identified with some classical constructions of stochastic calculus.

References in corpus (4)

• Renormalising SPDEs in regularity structures
• Geometric stochastic heat equations
• The motion of a random string
• On a new version of the Ito's formula for the stochastic heat equation