Set estimation from reflected Brownian motion

We study the problem of estimating a compact set $S\subset \mathbb{R}^d$ 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 based on tracking data. There are a variety of studies on the habitat of animals that employ the notion of home range. This paper offers theoretical foundations for a new methodology that, under fairly unrestrictive shape assumptions, allows one to find flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.