  We consider the estimation of the drift and the level sets of the stationary
distri- bution of a Brownian motion with drift, reflected in the boundary of a
compact set $S\subset R^d$ , departing from the observation of a trajectory of
this process. We obtain the uniform consistency and rates of convergence for
the proposed kernel based estimators. This problem has relevant applications in
ecology, in estimating the home-range and the core-area of an animal based on
tracking data. Recently, the problem of estimat- ing the domain of a reflected
Brownian motion was considered in Cholaquidis, et al. (2016), when the
stationary distribution is uniform and the estimation of the core-area, defined
as a level set of the stationary distribution, is meaningless. As a by-product
of our results, we obtain an estimation of the drift function. In order to
prove our re- sults, some new theoretical properties of the reflected Brownian
motion with drift are obtained, under fairly general assumptions. These
properties allow us to perform the estimation for flexible regions close to
reality. The theoretical findings are illustrated on simulated and real data
examples.
