On singular stochastic PDEs

2018/07/04 Wed 16:30 - 17:30
Room 110, Building No.3
Tadahiro Oh
The University of Edinburgh

In recent years, there has been significant progress on theoretical understanding of singular stochastic partial differential equations (SPDEs) with rough random forcing. The main difficulty in studying singular SPDEs lies in making sense of products of distributions, thus giving a precise meaning to an equation after appropriately modifying the equation (via renormalization).

In the field of stochastic parabolic PDEs, M. Hairer introduced the theory of regularity structures and gave a precise meaning to the so-called "subcritical" singular SPDEs such as the KPZ equation and the three-dimensional stochastic quantization equation, for which he was awarded a Fields medal in 2014. Around the same time, M. Gubinelli introduced the theory of paracontrolled distributions and solved a similar class of singular SPDEs. In this talk, I will first go over the basic difficulty in the subject and explain the main idea in these theories. Then, I will discuss recent developments in stochastic dispersive PDEs such as stochastic nonlinear wave and Schrödinger equations along with open problems in the field.