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Title: Rough PDEs and Hörmander's theorem for semilinear SPDEs
Author: Gerasimovics, Andris
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2019
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We investigate existence, uniqueness and regularity for local in time solutions of rough parabolic equations driven by a multiplicative noise of the form dut ́ Ltutdt “ Nputqdt ' ÿ d i“1 FiputqdXi t , where pLtqtPr0,Ts is a time-dependent family of unbounded operators acting on some scale of separable Banach spaces, while X ” pX, Xq is a two-step (non-necessarily geometric) rough path of Hölder regularity γ ą 1{3. Besides dealing with non-autonomous evolution equations, our results also allow for unbounded operations in the noise term (up to some critical loss of regularity depending on that of the rough path X). We later apply the machinery of rough paths to study the spectral properties of the Malliavin matrix of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander’s bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. A robust pathwise solution theory for such SPDEs allows us to use a pathwise version of Norris’s lemma to work directly on the Malliavin matrix, instead of the “reduced Malliavin matrix” which is not available in this context. On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Itô formula for rough PDEs. Finally, as a technical tool we introduce a version of the multiplicative sewing lemma, which allows to construct the so called product integrals in infinite dimensions. We later use it to construct a semigroup analogue for the non-autonomous linear PDEs as well as show how to deduce the semigroup version of the usual sewing lemma from it.
Supervisor: Hairer, Martin Sponsor: Leverhulme Trust
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral