Singular stochastic PDEs and related statistical physics models

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.

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.