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Title: Topics in singular SPDEs
Author: Schoenbauer, Philipp
ISNI:       0000 0004 9350 8873
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2020
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This thesis is concerned with problems arising in the field of singular stochastic partial differential equations (SPDEs). After two chapters providing an introduction to SPDEs in general and to the results of this thesis in particular, and a review of the preliminaries and notation needed, the main part of the thesis is contained in chapters 3 to 5. In chapter 3 we show a generalisation of the support theorem of Stroock and Varadhan to a large class of subcritical SPDEs. We prove that the topological support can be identified with the closure of the set of all solutions to a control problem associated to the SPDE. The main problem that we face here is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One immediate corollary of our general theorems, and one of the main contributions of this thesis, is a proof that the Phi43 measure in finite volume has full support and that the associated Langevin dynamic is exponentially ergodic. In the chapter 4 we establish Malliavin calculus for solutions to subcritical SPDEs. The main difficulty here is to show that Cameron-Martin functions can be lifted to renormalised models for the problem dependent regularity structure. Malliavin differentiability is a powerful tool to show existence of densities with respect to the Lebesgue measure, which we prove for certain finite-dimensional projections for a large class of equations. In the final chapter of this thesis we study the anisotropic KPZ equation in two dimensions. We identify a scaling such that smooth approximations given by introducing a hard-cutoff in Fourier space are tight, and we show that any subsequential limit is non-vanishing.
Supervisor: Hairer, Martin Sponsor: European Research Council
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral