Abstracts of the lectures
- Ivan Corwin : Integrable probability and the KPZ equation.
A large class of one dimensional stochastic particle systems are predicted to share the same universal long-time/large-scale behavior. By studying certain integrable models within this (Kardar-Parisi-Zhang) universality class we access what should be universal statistics and phenomena. In this series of lectures we focus mainly on two different integrable exclusion processes: q-TASEP and ASEP. Using them as a prompt, we will describe two approaches to studying exactly solvable particle systems: (1) Quantum many body systems and (2) Macdonald processes. These approaches unite integrability in various areas of probability including directed polymers, interacting particle systems, growth processes, and random matrix theory - Peter Friz: Rough paths and applications.
I intend to cover the following topics: The space of rough paths, Brownian motion as rough path, Integration against rough paths and relations to stochastic integration, Doob-Meyer type decomposition for rough paths, Rough differential equations and relations to Stochastic differential equations, Gaussian rough paths and integrability considerations. Applications to SPDEs. - Martin Hairer
Abstracts of the talks
- Yuri Bakhtin :
Burgers equation with random forcing in noncompact setting.
The Burgers equation is one of the basic nonlinear evolutionary PDEs. The study of ergodic properties of the Burgers equation with random forcing began in 1990's. The natural approach is based on the analysis of optimal paths in the random landscape generated by the random force potential. For a long time only compact cases of the Burgers dynamics on a circle or bounded interval were understood well. In this talk I will discuss the Burgers dynamics on the entire real line with no compactness or periodicity assumption on the random forcing. The resulting model belongs to the KPZ universality class. The main result is the description of the ergodic components and existence of a global attracting random solution in each component. The proof is based on ideas from the theory of first or last passage percolation. This is a joint work with Eric Cator and Kostya Khanin. - Marton Balasz : t^{1/3} scaling of fluctuations in asymmetric interacting systems.
In this talk I will briefly demonstrate a pretty general argument that gives time^1/3 scaling of current fluctuations in asymmetric particle systems. The method works with stochastic couplings and basic probabilistic arguments. The models themselves do not scale directly to the KPZ equation, but the results are close to those that give a better understanding of KPZ scaling too. I will also mention the technical point that restricts the proof and prevents us from applying the argument in full generality for a very wide class of systems. - Patrik Ferrari : Free energy fluctuations for directed polymers in 1+1 dimension.
The Kardar-Parisi-Zhang (KPZ) universality class includes directed polymers in random media in 1+1 dimension. According to the universality conjecture, for any finite temperature, the fluctuations of the free energy (e.g. for point-to-point) directed polymers is expected to be distributed as the (GUE) Tracy-Widom distribution in the limit of large system size. This distribution arose first in the context of random matrices. Detailed results as the fluctuation laws for models in the KPZ were, until recently, available only for "zero temperature models". We consider two models at positive temperature, a semi-discrete and the continuum directed polymer models, and determine the law of the free energy fluctuations. This talk is based on a joint work with Alexei Borodin and Ivan Corwin http://arxiv.org/abs/1204.1024 and their previous work http://arxiv.org/abs/1111.4408. - Massimiliano Gubinelli : Paracontrolled distributions
We use the notion of para-product to introduce a class of random generalised functions and a calculus of non-linear operations on them which allows us to understand few examples of singular random PDEs in a quite simple way. We will explain how to use these ideas to handle the KPZ equation, the stochastic quantization equation in 3 dimension and a parabolic Anderson model in two dimensions. - Tomohiro Sasamoto: The KPZ scaling functions for a two-component exclusion process.
The scaling functions which appear in the studies of models in the KPZ universality class are considered to be universal. The most typical example is the Tracy-Widom distributions which describe the fluctuations of the height in growth models or the current in driven lattice gases. Another example is the stationary two point function first studied by Pr"ahofer and Spohn.In a recent paper, H. Beijeren argued that the same universal scaling function would appear in more generic one-dimensional fluids. His arguments are based on mode-coupling approximation and have not been verified.
In this presentation, we consider a two-component exclusion process and discuss a possibility of observing the same scaling function. We explain the non-linear fluctuating hydrodynamical description of the process, the predictions based on the scheme and the knowledge on the exact stationary measure, and the comparison with the Monte-Carlo simulations.
The main purpose of the presentation is to explain the new developments in the area, provide some evidence (though numerical for now) and propose a possible direction for more rigorous investigations.
This is based on a collaboration with H. Spohn and P.L. Ferrari.
- Samy Tindel : Pathwise and Skorohod calculus in the plane
Recent advances have allowed to define an integration theory for a large class of Gaussian processes indexed by the plane, either by Malliavin calculus or rough paths methods. The aim of this talk is to compare the bidimensional integrals obtained with those two methods, computing explicit correction terms whenever possible. As a byproduct, we also give explicit forms of corrections in the respective change of variable formulas. Based on an ongoing work with Khalil Chouk.