Interface evolution in random environment

The main goal of this project is to develop mathematical methods for the mathematically rigorous analysis of the properties of interfaces evolving in a heterogeneous, random environment, described on a small scale by nonlinear PDEs with random coefficients.
It is motivated by the following situation: With an interface is associated a scalar quantity called its energy (think e.g. of its area) which it tries to decrease, i.e., it performs a gradient flow. This energy is perturbed through obstacles or impurities on a very small scale, and the system is driven by some large-scale force. The impurities are random, i.e., we have information only on the probability of finding certain impurities in a certain place, not on their precise nature and location. We are interested in the effective velocity and other qualitative properties of the interface on a large scale, much larger than the scale on which the perturbations vary. On that scale, the perturbations should “average out”, but we can ask the following natural questions:
What is the effective evolution law on a large scale, and what are the qualitative properties of the interface, e.g. on which scales does it look rough due to all the random heterogeneities? (The latter is related to the question of error estimates) How does all this depend on the law of the impurities?
The student will address, depending on his/her interest and background, some aspects of but not necessarily all these questions.