Set estimation from reflected Brownian motion

We study the problem of estimating a compact set S ⊂ Rd 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. The most common technique is to determine the minimum convex polygon (MCP), which is very restrictive regarding the shape of the habitat. Our approach allows us to find more and more flexible regions close to reality, as shown in some real data examples we provide