# Stochastic PDE's

Research in Stochastic Partial Differential Equations lies the interface of analysis [A] and probability theory [P]). For example, in trying to understand how solutions of macroscopic PDEs can be approximated by scaled particle systems at the microscopic level, one of the main questions is about the structure of the fluctuations of the approximating system around its hydrodynamic limit. This limit can often be described by an SPDE of linear type with a Gaussian solution. However if the space explored by the particle system lacks enough degrees of freedom, for example in the case of the one-dimensional asymmetric simple exclusion process, then the fluctuations become non-Gaussian and can only be described (if at all) by a non-linear SPDE. The so-called current fluctuations, crucial for the understanding of the non-linearity, are linked with the distribution of eigenvalues of random matrices. Other problems are the limiting behaviour of systems with multiple temporal and spatial scales, the numerical analysis of approximations to SPDEs [C] and the construction and analysis of MCMC methods in high dimensional spaces which provide a probabilistic tool for understanding statistical sampling techniques.

Researchers in this area:** **Assing, Hairer, Kolokoltsov, Roberts, Stuart, Tribe, Zygouras