Rough Stochastic Analysis Seminar

Abstract:  

The KPZ equation is a (1+1)-dimensional model of interface growth in the crossover regime. It is of interest both to the rough analysis community as an example of a singular stochastic PDE and also to the integrable probability community as a model featuring a rich structure that can be understood using the tools of integrable probability. Much of the integrable structure of the KPZ equation is accessed via taking limits of discrete models for which the structure is more transparent. In this talk, I will describe how the skew-symmetry of the KPZ equation under time-reversal provides a route to understanding some of the integrability properties of the KPZ equation directly in the continuum. This includes a proof of the invariant measure formula for the KPZ equation (joint work with Yu Gu and Tommaso Rosati) and a proof of the invariant measures for the multilayer and multicolor KPZ equations (joint work with Nicolas Perkowski, Ran Tao, and Hendrik Weber).

Abstract:  TBA

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: 

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.