Advanced Singular Stochastic PDEs
Global Estimates for singular SPDEs
A mini course deliverd for the summer school on Wave turbulence Summer school on Wave turbulence at MIT in July 2023.
Lecture 1: Examples of interesting equations, what is noise?, scaling
Lecture 2: Function spaces of distributions, Reconstruction Theorem, application to gPAM.
Lecture 3: Reconstruction for gPAM continued, interior regularity estimates.
Lecture 4: Stochastic estimates, global estimates for phi^4
List of References: pdf
Interacting Particle Systems and stochastic PDEs
An online course delivered for the workshop "Interacting Particle Systems and Hydrodynamic limits" at the Centre de Recherche Mathématiques.
Lecture 1: Introduction and Analysis of the Curie-Weiss and Kac-Ising model.
Lecture 2: Analysis of Kac-Ising, Introduction to phi^4 and renormalisation
Lecture 3: Strategy to prove convergence, convergence of trees, Wiener-Chaos decomposition
Lecture 4: More on trees and renormalisation, comments on implementation in particle case
An online course for graduate students of mathematics at BATH, BRISTOL, IMPERIAL COLLEGE LONDON, OXFORD AND WARWICK.
This is a second course on the topic and follows a TCC course I gave October - December 2020, as online mini-course for the Mathematics Department of Penn State University, June 2021 and during the Bath Symposium on PDE and Randomness . The slides for the previous lectures are below and recordings are available on request.
Lecturer: Hendrik Weber (University of Bath)
Lectures will take place Fridays 10-12am on Teams, from October 15th to December 3rd 2021.
If you are interested in participating, please register by sending a message to gradstud@maths.ox.ac.uk.
A number of important models from mathematical physics are given by stochastic PDEs, i.e. by partial differential equations which include a random term to describe some noise which is inherent to the system. A prominent example is the KPZ equation which describes the fluctuations around 1+1 dimensional moving interfaces, but there are many more.
Developing a mathematical theory for equations of this type was open for many years, the key difficulty being the ``roughness’’ of the natural noise terms. This irregularity forces the appearance of “infinite counter-terms” in the equation and makes standard solution methods inapplicable. Over the last 10 years this field has been revolutionized, most notably in works by Hairer, Gubinelli and their collaborators. A rigorous solution theory for many equations has been developed, and these solutions have in turn be applied e.g. to study scaling limits for microscopic particle models. On the mathematical level, these new methods constitute a beautiful mixture of ideas from several very different branches of mathematics, including the theory of rough paths, Gaussian analysis, PDE methods and perturbative field theory.
In these lectures I will extend the basic short time theory for the stochastic Anderson model that was developed last year and explain how it can be extended in several interesting ways. I will put a particular emphasis on the algebra of higher order expansions and how to describe it using the language of Hopf algebras. I will also discuss how to go beyond a short time solution theory and applications to study scaling limits of interacting particle systems.
Slides for Advances Singular Stochastic PDEs:
Lecture 1: Branched Rough paths 1
Lecture 2: Branched Rough paths 2
Lecture 3: Branched Rough paths 3
Lecture 4: Branched Rough paths 4
Lecture 5: Fixed point problem in Branched Rough paths. Finally PDEs
Lecture 6: The Regularity Structures formalism. Regularity structures, models, modelled distributions and reconstruction.
Lecture 7: The Regularity Structure for SHE
Lecture 8: Solving the fixed point problem for SHE
Slides for Penn State:
Lecture 1: Examples: KPZ, phi^4, multiplicative stochastic heat equation, Itô SDEs. A primer on controlled rough paths, sewing Lemma and construction of iterated stochastic integrals.
Lecture 2: White noise - definitions and properties. Measuring negative regularity of distributions. Regularity of white noise. Multiplication of functions and distributions - the Reconstruction Theorem 1.
Lecture 3: The Reconstruction Theorem 2. Linear Stochastic Heat Equation.
Lecture 4: The 2D continuuum Parabolic Anderson Model. Analysis of stochastic expansion. Renormalisation.
Lecture 5: Regularity Structures Vocabulary, Short-time solution theory for 2D-PAM via fixed point problem in space of modelled distributions, Rernormalisation again, Scope of the method and state of the art.
Slides for TCC 2020:
Lecture 1: Examples: KPZ, phi^4, Itô SDEs. A primer on controlled rough paths, sewing Lemma and construction of iterated stochastic integrals.
Lecture 2: More on rough paths, discussion of white noise, measuring regularity.
Lecture 3: Regularity of distributions, regularity of white noise, reconstruction theorem
Lecture 4: More on the reconstruction theorem
Lecture 5: The stochastic heat equation
Lecture 6: Iterated stochastic integrals and higher order expansions
Lecture 7: Discussion of Wick powers continued, Construction of phi^4_2 a la da Prato-Debussche, construction of 2D PAM.
Lecture 8: Complete construction of 2D PAM.