SPDEQFT: Stochastic PDEs meet QFT: Large deviations, Uhlenbeck compactness, and Yang-Mills
Lead Research Organisation:
University of Oxford
Department Name: Mathematical Institute
Abstract
The overarching goal of this proposal is
(a) to develop novel tools in the field of stochastic partial differential equations (SPDEs) and
(b) to apply them to mathematical quantum field theory (QFT), particularly quantum Yang-Mills (YM) theory.
The YM measure in the physical 4D space-time describes how elementary particles interact at the subatomic level. Its rigorous mathematical construction, however, has so far eluded substantial progress, and accordingly made the list of famously difficult "Millenium problems." On the other hand, SPDE theory has witnessed a number of recent breakthroughs, notably Hairer's theory of regularity structures, which has allowed to make sense of previously ill-posed, singular equations. This proposal aims to develop new tools in SPDEs and regularity structures to analyse the 2D YM measure. The research programme is structured into three projects:
1. We develop a solution theory for singular non-linear elliptic SPDEs. This significantly extends the scope of equations Hairer's theory allows to treat. At the same time, it provides the right framework to extend Uhlenbeck's compactness theorem to distributions. We use that generalisation to give a new gauge-fixed construction of the 2D YM measure, both on the optimal regularity space and with the natural gauge-invariant observables (Wilson loops) well-defined.
2. We show that singular (elliptic and parabolic) SPDEs can be analysed using classical Kusuoka-Stroock theory. This contributes to our theoretical understanding of regularity structures and, in particular, allows to derive precise Laplace asymptotics for these equations.
3. We prove precise Laplace asymptotics of the 2D YM measure in the low temperature limit. This is a novel insight into its qualitative behaviour and generalises a previous large deviation result, which has been obtained by completely different methods.
(a) to develop novel tools in the field of stochastic partial differential equations (SPDEs) and
(b) to apply them to mathematical quantum field theory (QFT), particularly quantum Yang-Mills (YM) theory.
The YM measure in the physical 4D space-time describes how elementary particles interact at the subatomic level. Its rigorous mathematical construction, however, has so far eluded substantial progress, and accordingly made the list of famously difficult "Millenium problems." On the other hand, SPDE theory has witnessed a number of recent breakthroughs, notably Hairer's theory of regularity structures, which has allowed to make sense of previously ill-posed, singular equations. This proposal aims to develop new tools in SPDEs and regularity structures to analyse the 2D YM measure. The research programme is structured into three projects:
1. We develop a solution theory for singular non-linear elliptic SPDEs. This significantly extends the scope of equations Hairer's theory allows to treat. At the same time, it provides the right framework to extend Uhlenbeck's compactness theorem to distributions. We use that generalisation to give a new gauge-fixed construction of the 2D YM measure, both on the optimal regularity space and with the natural gauge-invariant observables (Wilson loops) well-defined.
2. We show that singular (elliptic and parabolic) SPDEs can be analysed using classical Kusuoka-Stroock theory. This contributes to our theoretical understanding of regularity structures and, in particular, allows to derive precise Laplace asymptotics for these equations.
3. We prove precise Laplace asymptotics of the 2D YM measure in the low temperature limit. This is a novel insight into its qualitative behaviour and generalises a previous large deviation result, which has been obtained by completely different methods.
| Description | "A PDE construction of the 2D YM measure via Rough Uhlenbeck Compactness" - a joint project with Ilya Chevyrev and Abdulwahab Mohamed (both U Edinburgh & TU Berlin) |
| Organisation | Technical University Berlin |
| Department | Institute of Mathematics |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | All collaborators are contributing equally to this project. I (Tom Klose) have contributed my in-depth knowledge on rough paths theory and, in particular, the theory of regularity structures which I have been working on since 2017. |
| Collaborator Contribution | All collaborators are contributing equally to this project. Ilya Chevyrev is a leading expert on the "Stochastic Quantum Gauge Theories" and is leading an ERC Starting Grant on the same subject since 2023; he has been the original mentor for the MSCA Grant prior to his move to Europe. Abdulwahab Mohamed is a PhD student working with Ilya. Both have contributed important insights into the project, especially pertaining (but not limited) to the geometric aspects of the projects as well as the solution theory for (singular) elliptic (stochastic) PDEs. |
| Impact | This collaboration sits at the interface of mathematics and physics as it pertains to a topic of Constructive Quantum Field Theory; the preprint is in preparation. |
| Start Year | 2023 |
| Description | "A PDE construction of the 2D YM measure via Rough Uhlenbeck Compactness" - a joint project with Ilya Chevyrev and Abdulwahab Mohamed (both U Edinburgh & TU Berlin) |
| Organisation | University of Edinburgh |
| Department | School of Mathematics |
| Country | United Kingdom |
| Sector | Academic/University |
| PI Contribution | All collaborators are contributing equally to this project. I (Tom Klose) have contributed my in-depth knowledge on rough paths theory and, in particular, the theory of regularity structures which I have been working on since 2017. |
| Collaborator Contribution | All collaborators are contributing equally to this project. Ilya Chevyrev is a leading expert on the "Stochastic Quantum Gauge Theories" and is leading an ERC Starting Grant on the same subject since 2023; he has been the original mentor for the MSCA Grant prior to his move to Europe. Abdulwahab Mohamed is a PhD student working with Ilya. Both have contributed important insights into the project, especially pertaining (but not limited) to the geometric aspects of the projects as well as the solution theory for (singular) elliptic (stochastic) PDEs. |
| Impact | This collaboration sits at the interface of mathematics and physics as it pertains to a topic of Constructive Quantum Field Theory; the preprint is in preparation. |
| Start Year | 2023 |
| Description | "A Rough Breuer-Major Theorem" - a joint project with Nicolas Perkowski (FU Berlin) and Henri Elad Altman (Université Sorbonne Paris Nord) |
| Organisation | Free University of Berlin |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | All collaborators are contributing equally to this project. I (Tom Klose) am contributing to the project through my experience on rough paths theory as well as Gaussian analysis; I have been working on those topics throughout my doctoral dissertation. During a previous postdoctoral position at the University of Warwick, I have further strengthened my knowledge of Malliavin calculus and gained insights in this domain which are instrumental to this project. |
| Collaborator Contribution | All collaborators are contributing equally to this project. Nicolas Perkowski is a leading expert on singular stochastic PDEs, rough paths theory, and the theory of Markov processes; he has previously published in the field of "rough invariance principles". Henri Elad Altman is a specialist on Bessel stochastic PDEs and integration-by-parts formulae. Both have contributed important insights from their prior works in these domains. |
| Impact | This project is purely mathematical; the preprint is in preparation. |
| Start Year | 2024 |
| Description | "A Rough Breuer-Major Theorem" - a joint project with Nicolas Perkowski (FU Berlin) and Henri Elad Altman (Université Sorbonne Paris Nord) |
| Organisation | Sorbonne University |
| Country | France |
| Sector | Academic/University |
| PI Contribution | All collaborators are contributing equally to this project. I (Tom Klose) am contributing to the project through my experience on rough paths theory as well as Gaussian analysis; I have been working on those topics throughout my doctoral dissertation. During a previous postdoctoral position at the University of Warwick, I have further strengthened my knowledge of Malliavin calculus and gained insights in this domain which are instrumental to this project. |
| Collaborator Contribution | All collaborators are contributing equally to this project. Nicolas Perkowski is a leading expert on singular stochastic PDEs, rough paths theory, and the theory of Markov processes; he has previously published in the field of "rough invariance principles". Henri Elad Altman is a specialist on Bessel stochastic PDEs and integration-by-parts formulae. Both have contributed important insights from their prior works in these domains. |
| Impact | This project is purely mathematical; the preprint is in preparation. |
| Start Year | 2024 |
| Description | "Asymptotic Exit Problems for a Singular Stochastic Reaction-Diffusion Equation" - a joint project with Ioannis Gasteratos (TU Berlin) |
| Organisation | Technical University Berlin |
| Department | Institute of Mathematics |
| Country | Germany |
| Sector | Academic/University |
| PI Contribution | All authors are contributing equally to this project. I (Tom Klose) have previously worked on large deviations of stochastic PDEs, specifically those that are singular, and have contributed my expertise on the subject. In addition, this collaboration uses various results from my previous work on "Large Deviations of the \Phi^4_3 Measure via Stochastic Quantisation" (joint with A. Mayorcas, based at University of Bath) which has appeared in 2024. |
| Collaborator Contribution | All authors are contributing equally to this project. Ioannis Gasteratos has published various articles relating to large deviations, exit problems, and reaction-diffusion equations; he has contributed his expertise on all of these subjects. Specifically, this collaboration has benefited from Ioannis's previous work on "Uniform attraction and exit problems for stochastic damped wave equations" (joint with M. Salins and K. Spiliopoulos, both based at Boston University) which has appeared in 2025. |
| Impact | This project is purely mathematical in nature but sits at the interface of probability theory and mathematical physics, specifically statistical physics. |
| Start Year | 2024 |