19229411 Seminar on Stochastics "Anomalous Diffusions in Inhomogeneous Media"
- FU-Students only need to register via Campus Management.
- Non-FU-students are required to register via MyCampus/Whiteboard.
Summer Semester 2026
Lecturer: Dr. Guilherme de Lima Feltes
- Time and place: Tuesday, 12am--2pm, in SR 006, Königin-Luise-Str.24-26.
If you want to participate, please write an email to g.de.lima.feltes@fu-berlin.de
Prerequisites: Basic probability (Brownian motion, Markov processes), on the level of Stochastics I and II, and real analysis. Familiarity with Itô calculus is helpful. All advanced concepts will be introduced as needed.
Target group: Master students and BMS students.
Contents: A central goal of the seminar is to offer participants concrete entry points for Master’s thesis projects with several topics naturally extending into concrete research projects.
Diffusion in random or inhomogeneous media is a classical topic in probability and analysis, with applications ranging from materials science to fluid dynamics. In many situations, homogenisation theory explains why microscopic randomness leads, at large scales, to an effective diffusive behaviour. Mathematically this can be described by a Brownian motion with an effective diffusivity matrix, where the matrix is produced in the very non-linear form of averaging that is homogenisation. However, several important models exhibit anomalous diffusion: particles may spread more slowly (subdiffusion) or more rapidly (superdiffusion) than predicted by classical theory.
This seminar provides a self-contained introduction to such phenomena. We begin with classical stochastic homogenisation `a la Kipnis–Varadhan, including the H_{-1}–condition and invariance principles for diffusions in random environments. These results serve as a benchmark for understanding when and why homogenisation holds.
We then study canonical examples where homogenisation fails. On the subdiffusive side, we consider potential-type environments, such as Brox diffusion, where trapping mechanisms dominate. On the superdiffusive side, we focus on divergence-free environments, where circulation enhances transport. A central running example will be a particle moving in the curl of the two-dimensional Gaussian Free Field (GFF), which provides a clean and mathematically rich model for superdiffusive behaviour. This example will be examined from two very recent and complementary perspectives: via stochastic homogenisation of elliptic operators in divergence form, and via techniques inspired by mathematical physics, such as Fock space and resolvent expansions.
As a central theme, we will also investigate how a medium that itself evolves dynamically can affect the anomalous behaviour of the particle. In particular, we discuss models in which the environment is given by the stationary solution to a stochastic PDE. Recent results show that tuning the dynamics of the environment can restore diffusive behaviour or produce new anomalous regimes.
Discrete models (random walks in random environments) will be discussed alongside continuum limits to highlight universal mechanisms and methodological differences. Throughout, we compare classical tools with modern techniques developed to analyse anomalous scaling limits.
Talks
| Date | Ttitle | Speaker |
|---|---|---|
| 14.04. | Initial meeting, distribution of talks | Gui de Lima Feltes |
| 21.04. | From classical Homogenisation to super- and subdiffusion | Gui de Lima Feltes |
| 28.04. | From classical Homogenisation to super- and subdiffusion (cont.) | Gui de Lima Feltes |
| 05.05. | Some useful mathematical tools | Gui de Lima Feltes |
| 12.05. | General guidelines for preparation | Gui de Lima Feltes |
| 19.05. | Individual meetings to discuss progress | -- |
| 26.05. | Individual meetings to discuss progress | -- |
| 02.06. | Individual meetings to discuss progress | -- |
| 09.06. | An Introduction to Wiener Chaos decomposition of Gaussian Spaces | Balázs Kossovics |
| 16.06. | More on Wiener chaos and advanced tools in Homogenisation | Gui de Lima Feltes |
| 23.06. | A One-Dimensional Diffusion Process in a Wiener Medium | Festina Kika |
| 30.06. | The Gaussian free-field as a stream function: asymptotics of effective diffusivity in infra-red cut-off | Ivy Lynn Jay |
| 07.07. | √log -superdiffusivity for a Brownian particle in the curl of the 2d GFF | Balazs Kossovics |
| 14.07. | Brownian particle in the curl of 2-D stochastic heat equations | Maxim Blum |
Literature
- A list of references and literature will be provided.