  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.
