Set estimation from reflected Brownian motion

We study the problem of estimating a compact set S ⊂ R^d from a trajectory of a reflected Brownian motion in S with reflections on the boundary of S. We establish consistency and rates of convergence for various estimators of S and its boundary. This problem has relevant applications in ecology in estimating the home range of an animal on the basis of tracking data. There are a variety of studies on the habitat of animals that employ the notion of home range. The paper offers theoretical foundations for a new methodology that, under fairly unrestrictive shape assumptions, allows us to find flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.