An approximation theory for singular SPDEs with regularity structures.

In this talk, I will discuss the analytic aspects of an extension of the theory of regularity structures which accounts for a wider class of regularisations of the SPDE under consideration.

The theory of regularity structures originally developed by Hairer provides a solution theory for a wide range of classically ill-posed SPDEs by prescribing a way of renormalising a mollified version of the equation. The key ingredient of this approach is a generalised Taylor expansion of the solution dictated by the equation that is stable under relaxation of the mollification.

Our extension addresses and answers positively the question of whether the same solution theory can be recovered starting from other natural methods of regularising the original equation, such as spatial discretisations and viscous approximations. Our approach deals with all these approximations at once by assuming suitable small-scale bounds and re-running the whole programme in this general setting.

The resulting theory is expected to find applications in demonstrating invariance of measures, obtaining scaling limits and establishing weak universality results for singular SPDEs.