# Singular stochastic PDEs

**Lecturer:**Nicolas Perkowski**Date:**Wednesday, 11:15 - 12:45**Room:**A3 01**Language:**English**Target audience:**MSc students, PhD students, Postdocs**Keywords:**Stochastic partial differential equations, paracontrolled distributions**Prerequisites:**Basic knowledge of (continuous time) stochastic processes and functional analysis, in particular Schwartz distributions

## Abstract:

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

## Regular Lectures (Winter 2018/2019)

**Condition: the geometry of numerical algorithms**- Paul Breiding
- Tuesday, 09:00 - 10:00, MPI MiS G3 10, till January 30

**Information, Physics and Computation**- Pau Vilimelis Aceituno
- Wednesday, 13:00 - 14:30, MPI MiS A3 02

**Introduction to Random Algebraic Geometry**- Antonio Lerario
- Tuesday, 09:00 - 10:00, MPI MiS G3 10, from February 6 to March 27

**Random dynamical systems and stochastic porous media equations with nonlinear noise**- Benjamin Gess
- Monday, 16:15 - 17:45, MPI MiS A3 01

**Artiﬁcial Neural Networks and Machine Learning: Theoretical Foundations**- Nihat Ay
- Thursday, 11:15 - 12:45, MPI MiS A3 01

**Geometric and topological methods for the analysis of data in biology, neurobiology and other fields**- Jürgen Jost
- Friday, 13:30 - 15:00, MPI MiS A3 01

**Singular stochastic PDEs**- Nicolas Perkowski
- Wednesday, 11:15 - 12:45, MPI MiS A3 01

**Seminar on inverse spectral theory**- Angkana Rüland
- Thursdays, 9:15 - 10:45, Leipzig University, SG 3-14

**Introduction into regularity structures**- Falix Otto
- Wednesday, 09:10 - 11:00, MPI MiS A3 01

**Toric Varieties**- Mateusz Michalek
- Friday, 11:00 - 12:30, MPI MiS G3 10

**Morse Theory and Closed Geodesics**- Hans-Bert Rademacher
- Wednesday, 13:15 - 14:45, Leipzig University, SG 3-10

**An introduction to Information Geometry and Mathematical Population Genetics**- Tat Dat Tran
- Friday, 15:15 - 16:45, MPI MiS A3 02

**Reading Seminar: Representation theory - The basics**- Joscha Diehl
- first meeting: Tuesday 16.10.2018 at 11:00h, F3 Section

**Reading Seminar: Geometric Measure Theory**- Renan Assimos Martins, Kostas Zemas
- Monday, 15:00 - 17:00

**Geometric and Algebraic Combinatorics**- Bernd Sturmfels
- October, 29 – November 2 and November 16

**IMPRS-Ringvorlesung**- Nihat Ay, Christiane Görgen, Guido Montúfar, André Uschmajew, Stefan Hollands
- Tuesday, 10:00 - 12:00 (tba), MPI MiS G3 10