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arXiv · 1803.01744

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

Carlo Bellingeri
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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.

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Carlo Bellingeri. 2018-03-05. An It\^o type formula for the additive stochastic heat equation. https://doi.org/10.1214/19-ejp404

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