Noncommutative singular SPDEs
File(s)
Author(s)
Peev, Martin
Type
Thesis
Abstract
The focus of this dissertation is the study of the solution theory of noncommutative singular stochastic partial differential equations (SPDEs), motivated by the possible construction of non-Gaussian states on noncommutative algebras via stochastic quantisation.
The first chapter will give a short introduction to the topics discussed in this dissertation, in particular, relevant aspects of noncommutative probability theory, and the algebras we will be considering.
In Chapter 2, we deal with defining Hölder-Besov spaces with values in complete, locally convex, Hausdorff topological vector spaces. In particular, it shows that they can be represented as topological tensor products and how Hairer's reconstruction theorem, [Hai14], can be straightforwardly generalised to this setting.
In Chapter 3, we construct a localisation for Dirac Fermions using noncommutative points. We also give a detailed Da Prato-Debussche type argument for the local-in-time existence of solutions to the stochastic quantisation equation for the Higgs-Yukawa_2 model in a slightly regularised but still singular setting.
In Chapter 4, we introduce noncommutative regularity structures. Amongst other things, we provide a novel, fully inductive construction of negative renormalisation and a simplified proof of the cointeraction property. We use them to solve a number of noncommutative singular SPDEs. In particular, we do this in the q-Gaussian setting, where we define a new topology for these algebras, which allows us to prove intricate operator insertion estimates. We demonstrate the well-posedness of noncommutative rough differential equations, an It\^o formula, and the well-posedness of the Φ^4_3-equation in these algebras. Finally, we also finish showing the local-in-time existence of the Higgs--Yukawa_2 model without the regularisation.
The first chapter will give a short introduction to the topics discussed in this dissertation, in particular, relevant aspects of noncommutative probability theory, and the algebras we will be considering.
In Chapter 2, we deal with defining Hölder-Besov spaces with values in complete, locally convex, Hausdorff topological vector spaces. In particular, it shows that they can be represented as topological tensor products and how Hairer's reconstruction theorem, [Hai14], can be straightforwardly generalised to this setting.
In Chapter 3, we construct a localisation for Dirac Fermions using noncommutative points. We also give a detailed Da Prato-Debussche type argument for the local-in-time existence of solutions to the stochastic quantisation equation for the Higgs-Yukawa_2 model in a slightly regularised but still singular setting.
In Chapter 4, we introduce noncommutative regularity structures. Amongst other things, we provide a novel, fully inductive construction of negative renormalisation and a simplified proof of the cointeraction property. We use them to solve a number of noncommutative singular SPDEs. In particular, we do this in the q-Gaussian setting, where we define a new topology for these algebras, which allows us to prove intricate operator insertion estimates. We demonstrate the well-posedness of noncommutative rough differential equations, an It\^o formula, and the well-posedness of the Φ^4_3-equation in these algebras. Finally, we also finish showing the local-in-time existence of the Higgs--Yukawa_2 model without the regularisation.
Version
Open Access
Date Issued
2025-09-04
Date Awarded
2026-05-01
Copyright Statement
Attribution-NonCommercial-ShareAlike 4.0 International Licence (CC BY NC-SA)
Advisor
Chandra, Ajay
Hairer, Martin
Sponsor
Engineering and Physical Sciences Research Council
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)