Revisiting the Solution Theory for Singular SPDEs via Regularity Structures

The main aim of this talk is to present two refinements to the local solution theory for singular SPDEs provided by the framework of regularity structures; namely in expanding the allowed class of initial data and improving the topology in which convergence takes place as the ultraviolet cut-off is removed. As a pedagogical example to motivate these results, I will begin by sketching the strategy of proof for an at first sight unrelated result of Hairer, Ryser and Weber on the triviality of the Stochastic Allen-Cahn equation driven by space-time white noise in two spatial dimensions and then show how the refinements to the local solution theory mentioned above allow one to extend that result to the full subcritical regime.

The common feature of the aforementioned refinements is that they are based on the convergence of renormalised models in a stronger topology than typically appears in the literature. Following a recent line of work initiated by Linares, Otto, Tempelmayr and Tsatsoulis in the multi-index setting, we achieve the required convergence by appeal to the spectral gap inequality; in particular by applying tools introduced by Hairer and Steele in the tree-based setting. Time permitting, I will also discuss an independent ingredient in the generalisation of the work of Hairer, Ryser and Weber on the asymptotic behaviour of renormalisation constants.