Seminar Analysis and Mathematical Physics: Harprit Singh (Imperial College London)

Singular Stochastic PDEs

Singular stochastic partial differential equations (SPDEs) of the form 

∂_tu = △u + F (u, ∂_xu, ξ), 

where ξ is an irregular driving noise, arise in a variety of situations from quantum field theory to probability. After introducing some specific examples, we describe the main difficulty they share; they are singular due to the irregularity of the driving noise ξ.

In the first part of the talk we discuss a simple example where using the so-called “Da Prato-Debusche trick” is sufficient to deal with this difficulty. In the second half, we give a birds-eye view on how regularity structures provide a solution theory for such equations. In particular, we explain the role of subcriticality (super-renormalisability) and (half) Feynman diagrams in this theory. Lastly, we shall mention some recent results on the class of differential operators that are compatible with this general machinery and how this relates to the geometry of the underlying space.

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