Regularity Structures Winter Term 2026/27

Overview

This reading group is an introduction to Martin Hairer’s theory of regularity structures. It is aimed at people with little to no experience with stochastic PDEs, so no prior knowledge of singular SPDEs is assumed.

We will be closely following these lecture notes by Ajay Chandra and Hendrik Weber: Stochastic PDEs, Regularity Structures, and Interacting Particle Systems.

The first session will take place on Wednesday 7 October, 14:00–15:00, in Zeeman A1.01. We will then meet every Wednesday at the same time and place throughout Terms 1 and 2.

If you have any questions please email Haokai Xie or Theodoros Galanis.

Notes

We are putting together a set of notes based on the presentations given in the reading group, which are available under this link. After each session we will try to expand them with the material covered by the speaker.

Schedule

The following is a rough schedule for the reading group.

Session	Speaker	Topic	Corresponding sections of the notes
1	Theodoros Galanis	Overview: the equations, white noise, scaling, and why renormalisation is needed	§1, §1.1, §1.2
2	Haokai Xie	Parabolic Hölder–Besov spaces and a Kolmogorov theorem	§2.1
3	TBC	Linear theory: Duhamel’s formula, Schauder estimates and the graphical notation	§2.2
4	TBC	Nonlinear equations in d = 1, and a primer on iterated integrals	§2.3 (up to §2.3.1), Appendix A
5	TBC	Wick powers and the renormalised Picard iteration	§2.3.1, §2.3.2
6	TBC	The Da Prato–Debussche argument and its breakdown in d = 3	§3.1
7	TBC	Stochastic renormalisation of the Φ⁴₃ trees	§4.1 (first part), §4.2 (first part)
8	TBC	The abstract framework: regularity structures, models and modelled distributions	§3.2, §3.3, §3.4
9	TBC	Regularity structures for SPDEs: symbols, rules, and the Φ⁴₂ and Φ⁴₃ structures	§3.6, §3.7, §3.8
10	TBC	Canonical and renormalised models	§4.1, §4.2
11	TBC	Reconstruction and multiplication	§4.3
12	TBC	Integration	§4.4
13	TBC	The fixed point argument, and wrap-up	§4.5