# Singular stochastic PDEs and related statistical physics models

Lead Research Organisation:
University of Warwick

Department Name: Mathematics

### Abstract

The context of the proposal mainly concerns singular stochastic PDEs and related statistical physics models. By saying singular, we mean that the solution (or some of its derivatives) has wild oscillations with a frequency and magnitude blowing up to infinity at small scales. The singularities in the solutions to stochastic PDEs are typically almost everywhere. As a consequence, nonlinear operations of the solutions may not make sense as they take these high frequency oscillations into quantities that are typically infinity. Thus, the correct interpretation of the solutions to these equations usually requires renormalisation.

In the past three years, there have been major advances in the development of solution theories to a number of important singular SPDEs, including the three dimensional stochastic quantisation equation, the KPZ equation and the parabolic Anderson model in two and three dimensions. These equations are widely believed to be the universal models for the large scale behaviours of many systems in statistical mechanics. The successful construction of the solutions opens a way to study in detail these equations as well as the natural phenomena they represent. In this proposal, we aim to deepen the understanding of the quantitative behaviour of the solutions to these equations, and rigorously prove the universality phenomena for their related statistical physics models. We will also investigate how certain perturbations of the system (for example, asymmetry in phase coexistence models) can force its large scale behaviour to deviate from the expected universal limit.

In the past three years, there have been major advances in the development of solution theories to a number of important singular SPDEs, including the three dimensional stochastic quantisation equation, the KPZ equation and the parabolic Anderson model in two and three dimensions. These equations are widely believed to be the universal models for the large scale behaviours of many systems in statistical mechanics. The successful construction of the solutions opens a way to study in detail these equations as well as the natural phenomena they represent. In this proposal, we aim to deepen the understanding of the quantitative behaviour of the solutions to these equations, and rigorously prove the universality phenomena for their related statistical physics models. We will also investigate how certain perturbations of the system (for example, asymmetry in phase coexistence models) can force its large scale behaviour to deviate from the expected universal limit.

### Planned Impact

The work will have impact on a wide range of researchers (mainly probabilists, analysts, mathematical physicists, and statisticians) as well as the general public. In order for the results to reach these beneficiaries, I will do the following:

1. Present results at major international conferences as well as specialised research seminars;

2. Organise a workshop and invite a mixture of mathematicians, engineers and statisticians to speak;

3. Publish original research articles in the highest quality journals; also publish expository articles in non-specialised journals for general scientific community and the public;

4. Collaborate on various parts of the project with experts in the relevant subjects. These include both pure mathematicians (Peter Friz, Jean-Christophe Mourrat, etc.) and researchers from applied areas such as engineering (Nick Duffield).

5. Give public talks at local schools, and speak at departmental open days to high school students and their parents.

More details can be found in the Pathways to Impact document.

1. Present results at major international conferences as well as specialised research seminars;

2. Organise a workshop and invite a mixture of mathematicians, engineers and statisticians to speak;

3. Publish original research articles in the highest quality journals; also publish expository articles in non-specialised journals for general scientific community and the public;

4. Collaborate on various parts of the project with experts in the relevant subjects. These include both pure mathematicians (Peter Friz, Jean-Christophe Mourrat, etc.) and researchers from applied areas such as engineering (Nick Duffield).

5. Give public talks at local schools, and speak at departmental open days to high school students and their parents.

More details can be found in the Pathways to Impact document.

### Organisations

- University of Warwick, United Kingdom (Collaboration, Fellow, Lead Research Organisation)
- University of Oxford, United Kingdom (Collaboration)
- University College London, United Kingdom (Collaboration)
- University of Chicago, United States (Collaboration)
- Texas A&M University-Central Texas (Collaboration)
- Imperial College London, United Kingdom (Collaboration)
- Stanford University, United States (Collaboration)
- École normale supérieure de Lyon (ENS Lyon) (Collaboration)
- ENS de Lyon, France (Project Partner)
- Texas A & M University, United States (Project Partner)
- TU Berlin (Project Partner)

## People |
## ORCID iD |

Weijun Xu (Principal Investigator / Fellow) |

### Publications

Chang J
(2017)

*Signature inversion for monotone paths*in Electronic Communications in Probability
Fan C
(2019)

*Subcritical approximations to stochastic defocusing mass-critical nonlinear Schrödinger equation on R*in Journal of Differential Equations
Lyons T
(2017)

*Hyperbolic development and inversion of signature*in Journal of Functional Analysis
Mourrat J
(2017)

*From Particle Systems to Partial Differential Equations*
Xu W
(2018)

*Sharp Convergence of Nonlinear Functionals of a Class of Gaussian Random Fields.*in Communications in mathematics and statistics### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/N021568/1 | 01/09/2016 | 31/07/2018 | £261,659 | ||

EP/N021568/2 | Transfer | EP/N021568/1 | 15/10/2018 | 14/10/2020 | £185,034 |

Description | 1. In the preprint "Large-scale limit for interface fluctuation models" with Martin Hairer, we proved weak universality of the KPZ equations for a large class of interface growth models with general growth mechanisms beyond polynomials. This is a solid step forward to the complete understanding of weak universality phenomena. 2. In the publication "Signature inversion for monotone paths" with Chang, Duffield and Ni, we developed a simple and efficient algorithm to reconstruct any monotone path from its signature. This would have potential applications in the analysis of large monotone data sets. 3. In the preprint "Global well-posedness of mass critical stochastic nonlinear Schrodinger equation in d=1: small initial data" with Chenjie Fan, we gave the first proof of the global existence of solutions to critical nonlinear Schrodinger equation with stochastic inputs. This is the starting point for our future investigation in this field. |

Exploitation Route | I will try to publish the articles in leading journals, and present the results in international conferences. |

Sectors | Financial Services, and Management Consultancy,Transport,Other |

Description | 2D PAM moments |

Organisation | Stanford University |

Country | United States |

Sector | Academic/University |

PI Contribution | Joint mathematical paper. |

Collaborator Contribution | Joint mathematical paper. |

Impact | Moments of 2D parabolic Anderson model |

Start Year | 2016 |

Description | KPZ |

Organisation | Imperial College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Large-scale limit of interface fluctuation models, preprint in 2018. |

Start Year | 2015 |

Description | Phi^4 diagrams |

Organisation | University of Warwick |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Construction of $\Phi^4_3$ diagrams for pedestrians, published in Spring Proceedings in Mathematics and Statistics -- From Particle Systems to Partial Differential Equations (IV). |

Start Year | 2016 |

Description | Phi^4 diagrams |

Organisation | École normale supérieure de Lyon (ENS Lyon) |

Country | France |

Sector | Academic/University |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Construction of $\Phi^4_3$ diagrams for pedestrians, published in Spring Proceedings in Mathematics and Statistics -- From Particle Systems to Partial Differential Equations (IV). |

Start Year | 2016 |

Description | SNLS |

Organisation | University of Chicago |

Country | United States |

Sector | Academic/University |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Global well-posedness for the mass-critical stochastic nonlinear Schrödinger equation on R: small initial data, preprint 2018. |

Start Year | 2016 |

Description | Signature |

Organisation | Texas A&M University-Central Texas |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Signature inversion for monotone paths, published in Electronic Journal of Probability, Vol.22, 2017. |

Start Year | 2015 |

Description | Signature |

Organisation | University College London |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Signature inversion for monotone paths, published in Electronic Journal of Probability, Vol.22, 2017. |

Start Year | 2015 |

Description | Signature |

Organisation | University of Oxford |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Joint mathematical article. |

Collaborator Contribution | Joint mathematical article. |

Impact | Signature inversion for monotone paths, published in Electronic Journal of Probability, Vol.22, 2017. |

Start Year | 2015 |