Global Estimates for non-linear stochastic PDEs

Introduction

The project is concerned with the global behaviour of solutions to Stochastic Partial Differential Equations (SPDEs) from Mathematical Physics. These equations arise, for example, in the description of scaling limits of interacting particle systems and in the analysis of Quantum Field Theories.

Noise and Non-linearity

The equations contain noise terms that describe random fluctuations and act on all length scales. In this situation, the presence of a non-linear term can lead to divergencies. A subtle renormalisation procedure, which amounts to removing infinite terms, is needed.

Recent Developments

Over the last years, the understanding of non-linear SPDEs has been revolutionised, and a systematic treatment of the renormalisation procedure has been achieved. This led to a short-time well-posedness theory on compact domains for a large class of highly relevant semi-linear SPDEs.

Project Goals

In this project, I will describe the global behaviour of solutions of some of the most prominent examples, both in time and over infinite domains. This will be achieved by combining PDE techniques for the non-linear equations without noise and the improved understanding of the subtle small-scale stochastic cancellations.

Previous Work

I have already pioneered such a programme in an important special case, the dynamic Phi-4 model.

Specific Strands

The project has three specific strands:

1. Proving estimates for the stochastic quantisation equations of the Sine-Gordon and Liouville Quantum Gravity models and eventually Gauge theories, and providing a PDE-based approach to the celebrated 1-2-3 scaling of the KPZ equation.
2. Providing PDE-based constructions of Phi-4 models in fractional dimension and describing phase transitions in terms of mixing properties of the dynamics.
3. Treating degenerate parabolic equations and exploring if systems that fail to satisfy a fundamental "sub-criticality" scaling assumption can still be treated using SPDE techniques.