Many interesting applications from physics or biology, such as the understanding of transport processes in porous media or the evolution of populations, can be efficiently modeled using stochastic processes in a random medium. The occurrence of strong spatial or temporal inhomogeneities on microscopic scales is characteristic of random media. On macroscopic scales, i.e. On the other hand, a homogenization effect can typically be observed on length or time scales that are much larger compared to the microscopic inhomogeneities. This makes it possible to describe the dynamic behavior of the system through an effective stochastic process in a homogeneous, deterministic medium.
It is now of mathematical interest to understand the conditions under which homogenization occurs on the random medium and what relationship exists between the microscopic inhomogeneities on the one hand and the effective homogeneous quantities on the macroscopic level on the other.
- metastable behavior of Markov processes in disordered systems
- scale limits of stochastic processes in random media
- functional inequalities and their relationship to the long-term behavior of stochastic processes
- population dynamics
- Markov trials
- stochastic integration and stochastic differential equations
- metastability and potential theory