Rough PDEs and Hörmander's theorem for semilinear SPDEs
File(s)
Author(s)
Gerasimovics, Andris
Type
Thesis
Abstract
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.
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.
Version
Open Access
Date Issued
2019-10
Date Awarded
2020-02
Copyright Statement
Creative Commons Attribution NonCommercial Licence
Advisor
Hairer, Martin
Sponsor
Leverhulme Trust
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)