Rough Stochastic Analysis Seminar

Abstract: 

In regularity structures, the probabilistic input needed to solve singular stochastic PDEs is encoded by a random nonlinear object called the model, whose construction requires a suitable renormalisation.

This talk will focus on BPHZ-type theorems for regularity structures, which provide general constructions of renormalised models from natural classes of driving noises. I will describe some ideas that have led to a new understanding of such theorems, and discuss recent and ongoing work aimed at extending this picture beyond the settings in which the theory of regularity structures was first developed.

Based on joint works with Martin Hairer and with Lucas Broux and Harprit Singh, as well as ongoing projects with Martin Hairer, Xue-Mei Li and Francesco Pedullà, and with Máté Gerencsér and Yueh-Sheng Hsu 

Abstract: 

To construct the stochastic objects that are the building blocks of solutions of singular SPDEs, a common route is to control (and show the convergence of) the variances of their smooth approximations. In this talk we discuss what happens when these variances either diverge or vanish, necessitating a multiplicative renormalisation in the equation. Two main examples for such a situation are given by a subclass of super-critical SPDEs and the derivation of optimal convergence rates of approximations of solutions of "usual" singular SPDEs such as the KPZ equation. The talk is based on (partially ongoing) joint works with Konstantinos Dareiotis, Yueh-Sheng Hsu, Rhys Steele, and Fabio Toninelli.

Abstract: 

We will discuss how to study renormalisation of Stochastic PDEs in the presence of Boundary conditions, using the flow approach introduced by Pawel Duch. In particular we look at the Phi^4 equation with Dirichlet boundary conditions on a general domain with smooth boundary. We find the need for additional renormalisation accounting for the boundary conditions, as in previous work of Gerencer and Hairer. 

We also show that this boundary renormalisation can be taken constant even for domains with curved boundary. This is joint work with Majdouline Borji and Leonard Ferdinand. 

Abstract:

In this talk, I will show an approach to deriving a priori bounds for coercive SPDEs based on scaling. The basic idea is to first show bounds for the equation with a small noise and then rescale the bounds to a global scale. While many equations that the approach can handle have been treated recently with other methods, its advantages are that it is quite simple and allows one to state a single result that is applicable to a variety of equations, such as rough differential equations and parabolic/elliptic SPDEs. Based on joint work with Massimiliano Gubinelli.