Francesco Caravenna

Pavia-Bicocca-Indam Ph.D. program in Mathematics

PhD Course, Spring 2019

An Introduction to Regularity Structures

Francesco Caravenna

Overview

This course is an introduction to the theory of Regularity Structures, one of the greatest achievements of modern stochastic analysis, for which Martin Hairer was awarded the Fields Medal in 2014. This theory allows, for the first time, to give a robust meaning to important PDEs which are classically ill-posed due to the presence of singular terms, such as products of (possibly random) distributions of low regularity. At the heart of the theory lies a far-reaching generalization of the usual Taylor expansions, called modelled distributions, where ordinary monomials can be replaced by distributions of low regularity.

The course is thought not only for probability students, but also for students widely interested in analysis and (deterministic!) PDEs. The prerequisite is a basic knowledge in analysis (heat equation, distributions) and probability (Gaussian random variables, independence). Some familiarity with Brownian motion and stochastic integration is welcome, but not required.

Interested students are invited to contact me via email.

Lectures Log

• Lecture 1 (April 4). Examples of singular equations driven by white noise: KPZ, Φ⁴_d with d=2,3, α-SDE with 1/2 < α ≤ 1. Regularization of the noise and the need for renormalization: statement of the "Main Theorem". - Some key tools: distributions and Holder spaces with negative regularity; white noise and the convolution with the heat kernel. The Da Prato - Debussche argument for Φ⁴₂.
• Lecture 2 (April 11). Reminders on distributions. Convolution with smooth and non smooth functions. Scaled functions and the "Identification Lemma". Holder functions (of positive regularity) revisited. Holder distributions of negative regularity. Wavelets and characterization of Holder distributions.
• Lecture 3 (April 17). Distributions of finite order: link with negative Holder spaces via scaled functions. Elements of wavelets analysis and links with Holder distributions. The Reconstruction Theorem for a germ (= family of distributions) satisfying 1. bounded order and 2. coherence. Proof using wavelets.
• Lecture 4 (May 2). Definition of semi-norms for germs and continuity of the Reconstruction Map. An application of the Reconstruction Theorem: the product can be extended to a continuous bi-linear map: C^-a x C^b → C^-a provided b>a. Introduction to the axioms of Regularity Structures and their Models.
• Lecture 5 (May 9). Remarks on the non-density of smooth functions in the space C^b and on the extension of the product to a continuous bi-linear map: C^-a x C^b → C^-a for b>a. Axioms of Regularity Structures and their Models. Examples: the polynomial regularity structure; the rough path regularity structure.
• Lecture 6 (May 23). Reminders on the axioms of Regularity Structures and their Models. Definition of modelled distributions. The Reconstruction Theorem revisited. Notion of products in Regularity Structure.
• Lecture 7 (May 30). Reminders on modelled distributions. Topologies on models and modelled distributions, continuity of the Reconstruction Map. Example of a non-standard product. The rough path regularity structure: example of product; rough integral (via RT). The α-SDE with 1/2 < α ≤ 1 ("rough volatility"): strategy to lift the equation to modelled distribution; definition of the corresponding regularity structure. Introduction to the convolution with singular kernels.
• Lecture 8 (June 6). The α-SDE with 1/2 < α ≤ 1 ("rough volatility"). Definition of the regularity structure and of the "continuum model" for α > 1/4 (using suitable modifications of the Ito integral). Integration w.r.t. the singular kernel and multiplication with the noise at the level of modelled distributions. Solution of the "lifted" SDE as a fixed point in the space of modelled distributions, solved by contraction theorem for small times. Smooth approximations of the canonical model: renormalization is needed to have convergence.
• Lecture 9 (June 13). The α-SDE with 1/4 < α ≤ 1 ("rough volatility") revised. For any choice of "admissible" model (i.e. of the product zξ for z = K*ξ), the SDE can be lifted in the space of modelled distribution, where it has a unique solution (the product zξ is canonically defined for a class of z wide enough to find a solution). The canonical model from a smooth mollification ξ^ε of ξ does not converge to the continuum model defined by Ito integration: computation of the renormalization. - Introduction to the analysis of the Φ⁴₃ equation: symbols of the Regularity Structures, tree notation.
• Lecture 10 (June 25). Analysis of the Φ⁴₃ equation: recursive construction of the Regularity Structures (A,T,G) and canonical model (Π,Γ). Space of modelled distribution D^γ(V) with sector V relevant for the solution. Definition of product. Convolution operator K lifted to the space of modelled distribution (sketch). Lift of the Φ⁴₃ equation to the space of modelled distributions D^γ(V), determination of the exponent γ and relations among the coefficients. Non convergence of the canonical model: diverging expectation of Π(IΞ)². Elements of Wiener chaos theory and convergence of the renormalized Π(IΞ)² - E[Π(IΞ)²] (sketch).
  End of the course.

References

• Nils Berglund, An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures, Lecture Notes
• Ajay Chandra, Hendrik Weber, Stochastic PDEs, Regularity Structures, and Interacting Particle Systems, Lecture Notes
• Ivan Corwin, Hao Shen, Some recent progress in singular stochastic PDEs, Review Article
• Peter Friz, Martin Hairer, A Course on Rough Paths, Chapters 13-15, Springer (2014)
• Martin Hairer, Regularity structures and the dynamical Φ⁴₃ model, Review Article
• Martin Hairer, Introduction to regularity structures, Review Article
• Martin Hairer, A theory of regularity structures, Invent. Math. 198 (2014), 269-504
• Felix Otto, Singular SPDE with rough coefficients, Slides for RISM6 summer school
• Videos from a Workshop at the INI, Cambridge (3-7 Sep 2018) (with mini-courses by Felix Otto, Lorenzo Zambotti, Antti Kupiainen)

Where

Lectures will take place in room 3014, 3rd floor of the U5 building

Department of Mathematics, University of Milano-Bicocca, via Cozzi 55, Milano

When

Spring 2019, Thursdays 14:00-17:00, starting April 4, until mid June 2019.

The course will consist in about 8-10 lectures. Tentative schedule:

1. April 4
2. April 11
3. April 17
4. May 2
5. May 9
6. May 23
7. May 30
8. June 6
9. June 13 in room U9-07, U9 building (Viale dell'Innovazione, 10)
10. (NEW) June 25

What else?

For more information, do not hesitate to contact me via email.

Course details (PDF)