Nonlinear Stochastic Equations in Two and Three Dimensions

Stochastic partial differential equations (SPDE) describe the behaviour of spatially extended systems under the influence of noise. They arise naturally in various fields of applications as diverse as data mining, mathematical finance, and population dynamics and genetics.

The present proposal aims to study a class of stochastic partial differential equations from statistical mechanics. Many particle models exhibit a behaviour called phase transition, where the behaviour of the system changes drastically when one changes a given system parameter beyond a critical point. It is a very exciting question to understand the behaviour of such a system near a critical point.

In such a regime one expects the dynamics to be governed by a non-linear SPDE. Analytically the understanding of these equations is very challenging, because of the interaction between the rough noise term and the non-linear evolution. But this is also what gives rise to interesting phenomena.

In this proposal, we aim to deepen the understanding of these equations. On the one hand we will study the behaviour under the influence of a small noise term. Then we will establish that two different kinds of particle models can indeed be described by such an SPDE.

The main beneficiaries of the proposed research will be:

- Research mathematicians working on SPDEs and particle systems. They will profit directly from the new techniques developed in the proposed research.
- The proposal aims to have an impact on a broader group of mathematicians, in particular, researchers working on deterministic PDEs as well mathematical physicists.
- On a longer timescale, advanced techniques for SPDEs will have an impact on applied sciences, for example in Statistics.
- Graduate students from the relevant areas will profit directly from interactions with the PI.

In order to reach these beneficiaries the PI will do the following:

- The work will be presented at key international conferences, such as the 38th Conference on Stochastic Processes and their Applications (Oxford, 07/15) and the 18th International Congress on Mathematical Physics (Santiago de Chile, 08/15).
- The PI will present his results at smaller, more focused workshops and seminars, in the UK and Europe. One of these will be the workshop on ``Stochastic Analysis: Around the KPZ Universality" at the MFO in 06/14 to which the PI has been invited .
-The PI will organise an international workshop on SPDEs and particle systems, with up to 20 invited speakers, at the University of Warwick.
- To ensure timely availability of the project's results to other academics working in the area, the PI will post these on various preprint servers, including his own home page, and also on arXiv.
- The work will be published in high quality journals, such as Communications on Pure and Applied Mathematics, Probability Theory and Related Fields, and Communications in Mathematical Physics.
- The PI will teach a course in the EPSRC Taught Courses Centre on related material.
- The PI will continue his collaboration with Prof. Pinski and Prof. Stuart on application of methods from stochastic analysis to molecular dynamics. This will accelerate the impact of advanced techniques from stochastic analysis to the applied sciences.

The PI will collaborate on various sections of the project with Prof. Martin Hairer (Warwick), Prof. Thierry Bodineau (Paris) and Dr. Jean-Christophe Mourrat (Lyon). The involvement of these established scientists, will increase the prospect of the project yielding high-impact results.