Renormalisation and a priori bounds for singular SPDEs
File(s)
Author(s)
Steele, Rhys
Type
Thesis
Abstract
This thesis pertains to topics in the application of the theory of regularity structures to singular stochastic partial differential equations (singular SPDEs) with a focus on problems motivated by stochastic quantisation of the Phi_3^4 field theory.
The first set of original work presented in this thesis pertains to the construction of the renormalised models that are required as input for the analytic solution theory for singular SPDEs in the framework of regularity structures. We provide a new proof of convergence of the BPHZ renormalised models as regularisation of the noise is removed. This proof is a significant simplification of the work that is currently available in the literature.
As an underlying probabilistic assumption, we assume that the noise satisfies a spectral gap inequality that allows us to leverage the recursive nature of the construction of suitable regularity structures of decorated trees.
This work entails the development of a number of variants of the spaces of modelled distributions that have previously appeared in the literature that track finer properties of the corresponding distributions. We then show that this framework can be used to describe the Fréchet derivative of a renormalised model with respect to the driving noise and thus leverage these tools to provide a proof of the required stochastic estimates.
Additionally, in a second area of original work, we provide a simple argument that shows that the Phi_3^4 measure has quartic exponential tails, as one would expect from the formal expression for the measure. This provides what are to our knowledge the best available bounds for the measure in the current literature. Our bounds are sufficient to see that the moment problem for the Phi_3^4 measure is well-posed and provide a particularly simple path to observing the non-Gaussianity of that measure.
The first set of original work presented in this thesis pertains to the construction of the renormalised models that are required as input for the analytic solution theory for singular SPDEs in the framework of regularity structures. We provide a new proof of convergence of the BPHZ renormalised models as regularisation of the noise is removed. This proof is a significant simplification of the work that is currently available in the literature.
As an underlying probabilistic assumption, we assume that the noise satisfies a spectral gap inequality that allows us to leverage the recursive nature of the construction of suitable regularity structures of decorated trees.
This work entails the development of a number of variants of the spaces of modelled distributions that have previously appeared in the literature that track finer properties of the corresponding distributions. We then show that this framework can be used to describe the Fréchet derivative of a renormalised model with respect to the driving noise and thus leverage these tools to provide a proof of the required stochastic estimates.
Additionally, in a second area of original work, we provide a simple argument that shows that the Phi_3^4 measure has quartic exponential tails, as one would expect from the formal expression for the measure. This provides what are to our knowledge the best available bounds for the measure in the current literature. Our bounds are sufficient to see that the moment problem for the Phi_3^4 measure is well-posed and provide a particularly simple path to observing the non-Gaussianity of that measure.
Version
Open Access
Date Issued
2022-09-27
Date Awarded
01/03/2023
License URL
Advisor
Hairer, Martin
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)