  Consider an multidimensional obliquely reflected Brownian motion in the
positive orthant, or, more generally, in a convex polyhedral cone. We find
sufficient conditions for existence of a stationary distribution and
convergence to this distribution at the exponential rate, as time goes to
infinity. We also prove that certain exponential moments for this distribution
are finite, thus providing a tail estimate for this distribution. Finally, we
apply these results to systems of rank-based competing Brownian particles.
