Quantitative homogenization of forced geometric motions through random fields of obstacles

Consider the evolution of sets by forced mean curvature flow through a field of random obstacles. The effective large scale behaviour is expected to be a first order motion. However, previous results heavily relied on the assumption that there is a global minimum speed of expansion and hence on the absence of any actual obstacles.

We obtain a quantitative homogenization result even with impenetrable obstacles, potentially allowing the interface to get stuck locally, eventually leading to enclosures behind a main front. So far in this regime not even a qualitative stochastic homogenization result had been available. The existence of a global minimum speed is replaced with a probabilistic assumption using the notions of "approximate stability" and an "effective minimum speed". This assumption is satisfied for example if the obstacles are distributed according to a Poisson point process with low enough intensity. The talk is based on joint work with Julian Fischer (Institute of Science and Technology Austria).