Summer school: PDE and probability

Contribution List

7. Anomalous diffusion and Euler flows

Prof. Jan Burczak (Universität Leipzig)

Scalar anomalous dissipation is a phenomenon in fluid dynamics in which a quantity advected by a fluid (e.g. the density of pollutant particles advected by a water current) is dissipated even 'without the final assistance of viscosity'. This behaviour has strong links to turbulence: in particular to Kolmogorov's 'zeroth law', to Onsager's theory, and to Richardson's hypothesis of turbulent...

6. Large-scale behaviour of random motions with long memory

Balint Toth (Bristol & Budapest)

The course will discuss at least two of the following three topics.

1.
Random walks and diffusions in divergence-free random drift fields.
Motivation, examples, main questions.
Diffusive limit (CLT) and homogenisation under H_{-1} and ellipticity. (In probability and quenched, w.r.t. the environment)
Relaxing the ellipticity condition. Motivation (why care?). Diffusive limit and...

3. Quantum diffusion and random matrix theory

Felipe Hernandez (Penn State University)

The random Schrodinger equation and its discrete analogue, the tight-binding model, have been studied as models of electron transport in disordered systems. It is conjectured that the long-time behavior is either localized or diffusive. In the localized regime, solutions remain trapped near their initial position. This phase has been established rigorously near the spectral edges, and is...

4. Rigorous justification of wave kinetic theory

Zaher Hani (University of Michigan)

Wave kinetic theory, also known as wave turbulence theory, is the kinetic description of systems with an asymptotically large number of weakly interacting waves. It is the wave analog of Boltzmann’s kinetic theory for particle systems, and plays a fundamental role in various fields of science. Over the last five years, in collaboration with Yu Deng (Chicago), we have developed a robust...

5. Statistical properties of a hard sphere gas

Thierry Bodineau (IHES & CNRS)

A gas dynamics can be modeled by a billiard of hard spheres, moving according to the laws of classical mechanics. Initially the spheres are randomly distributed according to a probability measure which is then transported by the deterministic flow of the Hamiltonian dynamics. Since the seminal work of Lanford, it is known that the gas density converges, in the low density limit, towards a...

1. TBA

Prof. Martin Hairer (EPFL)

2. TBA

Prof. Sourav Chatterjee (Stanford University)