Abstract:

Many applications, such as porous media or composite materials, involve heterogeneous media which are modeled by random fields. These media are locally irregular but are "statistically homogeneous" in the sense that their law has homogeneity properties. Considering random motions in such a random medium, it turns out often that they can be described by their effective behaviour. This means that there is a deterministic medium, the effective medium, whose properties are close to the random medium, when measured on long space-time scales. In other words, the local irregularities of the random medium average out over large space-time scales, and the random motion is characterized by the "macroscopic" parameters of the effective medium. How do the macroscopic parameters depend on the law of the random medium?

As an example, we consider the effective diffusivity (i.e. the covariance matrix in the central limit theorem) of a random walk among random conductances. It is interesting and non-trivial to describe this diffusivity in terms of the law of the conductances. The Einstein relates this diffusivity with the derivative of the speed of a biased random walk among random conductances. We explain the Einstein relation and we also discuss monotonicity questions for the speed of a biased random walk among random conductances.

The talk is based on joint work (in progress) with Noam Berger, Xiaoqin Guo and and Jan Nagel.