These (work-in-progress) lecture notes are associated to an ongoing course taught at the Max Planck Institute for Mathematics in the Sciences.
Lecture 1: Examples of Singular SPDEs, Definition of Hölder-Besov spaces
Lecture 2: Hölder regularity of White Noise, Young's product Theorem and Coherent Germs
These lecture notes are associated to a course on regularity structures taught at the Max Planck Insitute for Mathematics in the Sciences in 2024. They have not been updated since each of the lectures was taught and therefore still exist only in an unpolished format.
Lecture 1 First examples, White noise, Holder-Besov spaces, Young Products (Statement)
Lecture 2 Proof of the if direction in the Young Products statement, a first Reconstruction Theorem
Lecture 3 Recap of Young Products and Reconstruction, Elliptic Phi_4^4 by the Da Prato--Debussche Trick
Lecture 4 Wick Renormalisation for the Elliptic Phi_4^4 Equation
Lecture 5 The Role of Symmetries in Renormalisation, Heuristic Discussion of the Elliptic Phi_5^4 Equation
Lecture 6 Regularity Structures and Models
Lecture 7 Examples of Models, Definition of Modelled Distributions and statement of a corresponding Reconstruction Theorem
Lecture 8 Proof of the Reconstruction Theorem for Modelled Distributions, Products of Modelled Distributions
Lecture 9 Abstract integration maps, Admissible models, a Pointed Schauder Estimate
Lecture 10 Proof of Multilevel Schauder Estimates
Lecture 11 An Analytic Solution Theory for Elliptic Phi_5^4, a first brief discussion of renormalisation
Lecture 12 Regularity Structures Determined by Rules, A first step to algebraising the structure group
Lecture 13 Algebras, Coalgebras, Bialgebras, Hopf Algebras and their character group.
Lecture 14 Construction of Antipodes, Comodules, Models from preparation maps
Lecture 15 (slides) Overview of Stochastic Estimates for Models via the Spectral Gap Inequality
The early lectures of this course were accompanied by some problem sheets.
Sheet 1 Properties of Holder-Besov Spaces
Sheet 2 Compactly supported test functions with the semigroup-like property, the existence part of the Young Product statement
Sheet 3 Schauder Estimates, Properties of Germs, Small Parameter Well-Posedness of Elliptic Phi_4^4
Sheet 4 Renormalisation for Elliptic Phi_4^4 via the Spectral Gap Inequality
For the 2024 lecture course on regularity structures, I wrote the following very short primer on distributions: A Short Primer on Distributions