Topics in singular SPDEs
File(s)
Author(s)
Schoenbauer, Philipp
Type
Thesis
Abstract
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.
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.
Version
Open Access
Date Issued
2019-09
Date Awarded
2020-01
Copyright Statement
Creative Commons Attribution NonCommercial Licence
Advisor
Hairer, Martin
Sponsor
European Research Council
Grant Number
Project 615897
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)