Effective properties of interface evolution in a random environment

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This is mathematical research connecting the analysis of partial differential equations (PDEs) and (applied) probability.The main goal of this project is to derive macroscopic evolution laws for interfaces in heterogeneous, random environments, described on a small scale by nonlinear PDEs with random coefficients. In particular, we want to go beyond classical homogenisation in order to treat systems where a long-range collective behaviour emerges. A key aspect of this project is to establish a new link between analysis and probability, benefitting both fields, by working on problems motivated by applications.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 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 the questions is: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? How does all this depend on the law of the impurities?This is important because we are interested in the reaction of a system on the scale of our everyday life to an input on that scale. E.g. we would like to know how a piece of metal changes shape in a car crash, we are not interested in the position of each single atom, and we wouldn't be able to compute those anyway. But most realistic materials have some random structure on a fine scale.