## Bath Mathematical Symposium on

# "PDE and Randomness"

### PDE and Randomness symposium, 1st - 10th of September 2021, University of Bath, UK.

The last few years have witnessed a number of exciting developments at the interface of Probability theory and the theory of PDE. Among these are

Singular stochastic PDEs: starting with the discovery of regularity structures / paracontrolled distributions a systematic theory of white noise driven SPDEs from mathematical Physics was developed.

New results on the large-scale behaviour of discrete models from Statistical Mechanics have emerged.

Stochastic Homogenization: the classical theory of stochastic homogenization has been completely rewritten and quantified.

Dispersive PDEs with random data have been shown to display a much better solution theory than naive regularity considerations would suggest.

These a priori quite different topics share a number of common features, among them the question of renormalization / removal of infinite terms as well as the prominent use of tools from regularity theory / harmonic analysis.

The aim of this symposium is to bring together some of the leading protagonists in these discoveries, to showcase their results, similarities and differences in between them and to prepare the grounds for future developments.

*Due to the spread of the Delta variant in UK both school and the workshop will be held online. Information about the online registration will appear shortly.*

# Workshop

## 6th of September - 10th of September.

## List of speakers:

**Roland Bauerschmidt***Cambridge*) -**Slides****Thierry Bodineau**(École Polytechnique) -**Slides****Giuseppe Cannizzaro**(Warwick) -**Slides****Ajay Chandra**(Imperial College London)**Ilya Chevyrev**(University of Edinburgh) -**Slides****Benjamin Fehrman**(Oxford) -**Slides****Julian Fischer**(IST Austria)**Benjamin Gess**(Bielefeld University; Max Planck Institute, Leipzig) -**Slides****Yu Gu**(CMU, Pittsburgh)**Martin Hairer****Slides****Milton Jara**(IMPA, Rio de Janeiro) -**Slides****Roman Kotecký**(University of Warwick)**Antti Kupiainen**(University of Helsinki) -**Sli****des****Jean-Christophe Mourrat**(ENS Lyon and CNRS)

**Andrea Nahmod**(University of Massachusetts Amherst) -**Slides****Tadahiro Oh**(University of Edinburgh) -**Slides****Felix Otto**(MPI, Leipzig) -**Slides****Leonardo Tolomeo**(HCM, Bonn) -**Slides****Fabio Toninelli**(TU Vienna) -**Slides****Balint Toth**(University of Bristol; Alfréd Rényi Institute of Mathematics**,**Budapest) -**Slides****Pavlos Tsatsoulis**(MPI, Leipzig) -**Slides****Nikolay Tzvetkov**(Cergy-Pontoise) -**Slides****Rongchan Zhu**(Bielefeld University; Beijing Institute of Technology) -**Slides****Xiangchan Zhu**(Bielefeld University; Chinese Academy of Sciences, Beijing) -**Slides****Nikos Zygouras**(University of Warwick) -**Slides**

*Schedule of the Workshop (6th - 10th of September):* **SCHEDULE**

*Titles and Abstracts:* **TITLES****&ABSTRACTS**

Zoom link for the workshop will be sent on the 5th of September. If you did not receive it by the end of the day please email to **pderandom@bath.ac.uk**.

# Summer School

## 1st - 3rd of September.

## The summer school will consist of three courses:

**Abstract**

Introduction to Stochastic Homogenization.

Besides being of mathematical interest, PDEs with random coefficient field appear naturally in a number of applications, for instance as models for materials with random heterogeneities. On large scales, such PDEs with random coefficient field often behave like a PDE with constant coefficients. The field of stochastic homogenization - the derivation of such effective macroscopic descriptions for PDEs with microscopically random coefficient fields - has seen a series of breakthroughs in the last decade.

In this lecture series, we give an introduction to the Gloria-Otto approach to quantitative stochastic homogenization. We start with a short introduction to homogenization at the example of linear elliptic PDEs with periodic coefficient field, introducing basic notions of homogenization theory. Subsequently, we give a brief introduction to concentration inequalities. In the main part, we illustrate how concentration inequalities and elliptic regularity estimates play together to provide quantitative error estimates in stochastic homogenization of linear elliptic equations.

**Abstract**

Non-equilibrium stationary states (NESS) are stationary states of Markov chains that are characterized by the presence of non-zero currents. From a probabilistic point of view, these are stationary, non-reversible states. Unless some combinatorial miracle happens, these states can not be described explicitly. The aim of this course is to explain how entropy methods can be used in order to derive law of large numbers and central limit theorems for a family of NESS arising from driven diffusive systems. This provides a first step into a general strategy to study more general NESS.

**Abstract**

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.

The aim of this course is to give an introduction to some of these ideas and to show how they work together.

*To request videos of the lectures please write to ***ag2616@bath.ac.uk***.*

### Please register by completing the following form.

Please note that the google form does not send an automatic reply that you had been registered.

Registration for the school will be closed on 30th of August 23.59 London time.

Registration for the workshop will be closed on 4th of September 23.59 London time.

If you wish to be put on the mailing list after the registration is closed please email to **pderandom@bath.ac.uk**

specifying which event you want to be registered for.

Organizers:

Professor Hendrik Weber

Dr Andris Gerasimovics

For any additional information and queries please email on: **pderandom@bath.ac.uk**

Please be aware of the possible scams! We or anyone on our behalf will never call you to ask for your bank details or some other valuable information. All communications will be done through the above email.

### Location: Online

Funding is provided by The Royal Society.