Francesco Pedullà (Maths Research Postgraduate, Imperial)
Title
An asymptotic upper bound on renormalisation constants with regularity structures
Abstract
When studying the properties of renormalised solutions to singular SPDEs, a helpful tool can be relatively precise descriptions of the asymptotic behaviour of the renormalisation constants that appear in the equation as the ultraviolet cut-off is removed.
The aim of this talk is to present a simple inductive argument to obtain asymptotic upper bounds on the renormalisation constants arising in the theory of regularity structure. These bounds are expected to be sharp except in the presence of logarithmically divergent counter-terms and essentially only assume the stochastic estimates used to solve the equation.
As an application, we extend the result of Hairer, Ryser and Weber on the triviality of the solution to the dynamical Phi42 equation without renormalisation to the full sub-critical regime. This is achieved by combining our asymptotic bounds and an extension of the existing solution theory for singular SPDEs provided in the regularity structures literature, which is aimed at allowing for a wider class of initial conditions.
This is joint work with Rhys Steele.
Martin Peeve (Maths Research Postgraduate, Imperial)
Title
Renormalising Non-Commutative Singular PDEs
Abstract
When attempting to construct QFTs that include Fermions using the methods of Stochastic Quantisation, one is naturally forced to consider noncommutative stochastic PDEs. I shall show how to formulate SPDEs driven by noncommutative noises in terms of algebra-valued singular PDEs. Furthermore, I will describe how one can renormalise the singular products appearing in such equations for a set of algebras – including free probability – interpolating between Fermions and Bosons by appropriately modifying their topologies. This talk will be based on joint work with Ajay Chandra and Martin Hairer.