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Itô formulae for the stochastic heat equation via the theories of regularity structure and rough paths

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, in order 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 : Thursday, September 10, 2020 - 5:20:00 PM
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Carlo Bellingeri. Itô formulae for the stochastic heat equation via the theories of regularity structure and rough paths. Probability [math.PR]. Sorbonne Université, 2019. English. ⟨NNT : 2019SORUS028⟩. ⟨tel-02935889v2⟩



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