Ito formulae for the stochastic heat equation via the theories of rough paths and regularity structures

Abstract : In this thesis, we develop a general theory to prove the existence of several Itô formulae on the one-dimensional stochastic heat equation driven by additive space-time white noise. That is denoting by u the solution of this SPDE for any smooth enough function f we define some new notions of stochastic integrals defined upon u, which cannot be defined classically, to deduce new identities involving f(u) and some non-trivial corrections. These new relations are obtained by using the theory of regularity structures and the theory of rough paths. In the first chapter, we obtain a differential and an integral identity involving the reconstruction of some modelled distributions. Then we discuss a general change of variable formula over any Hölder continuous path in the context of rough paths, relating it to the notion of quasi-shuffle algebras and the family of so-called quasi-geometric rough paths. Finally, we apply the general results on quasi-geometric rough paths to the time evolution of u. Using the Gaussian behaviour of the process $u$, most of the terms involved in these equations are also identified with some classical constructions of stochastic calculus.
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Submitted on : Sunday, August 18, 2019 - 10:49:06 PM
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Carlo Bellingeri. Ito formulae for the stochastic heat equation via the theories of rough paths and regularity structures. Probability [math.PR]. Sorbonne Université, 2019. English. ⟨tel-02267116⟩

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