Singular stochastic PDEs

Recent years have shown tremendous progress in our understanding of "singular stochastic PDEs", which are ill posed due to the interplay of very irregular noise and nonlinearities which may lead to resonances that have to be removed by a renormalization procedure. In fact all white-noise driven parabolic SPDEs in dimensions larger than \(d=1\) are singular. To solve these equations it was necessary to come up with new, analytic rather than stochastic, notions of solutions, for example with Hairer's regularity structures or with paracontrolled distributions developed by myself together with Gubinelli and Imkeller. In the lecture I will give an introduction to paracontrolled distributions and applications and I will try to communicate the main ideas without getting lost in technicalities. After introducing the basic tools needed to solve singular SPDEs I will focus on applications like

• invariance principles
• constructing domains for singular operators (Anderson Hamiltonian, infinitesimal generators of diffusions with distributional drift)
• descriptions of singular SPDEs in terms of singular diffusions ("Feynman-Kac formula")
• Barashkov-Gubinelli's variational approach to constructing Gibbs measures like \(\Phi^4_3\)
• aspects of the large scale behavior of some singular SPDEs

Date and time info
Wednesday, 11:15 - 12:45

Keywords
Stochastic partial differential equations, paracontrolled distributions

Prerequisites
Basic knowledge of (continuous time) stochastic processes and functional analysis, in particular Schwartz distributions

Audience
MSc students, PhD students, Postdocs

Language
English