My current research and main topic of my thesis, is centred around the study of various non-local singular SPDEs. This work is inspired by recent developements in powerful techniques for studying singular SPDEs in general, those being the theories of Paracontrolled Distributions and Regularity Structures. So far a lot of work has been done on extending these short-time results to global or blow-up results for SPDEs with local non-linearities; e.g Phi^4_3, KPZ and PAM. One major aim of my project is to investigate the possibillity of extending these results to singular SPDEs with non-local interactions.

A second area I am interested in is in a sense the reverse of these questions; what positive effect can singular noise have on otherwise ill-posed or badly behaved non-local systems. An example of such a result was shown by Gumbinelli et.al in 2010 where they showed that the addition of noise to the Euler point vortex dynamics renders the system of SDEs well-posed for all initial data - an improvement on almost all initial data as is true for the deterministic system. Noise was also shown to have a positive effect on the continuation of point charge dynamics governed by the Vlasov-Poisson-Fokker-Planck equations in 1D. I am interested in studying the possibillities of similar affects on other more singular equations, or those displaying different singular phenomena.