Reflecting Brownian motion in three dimensions: a new proof of sufficient conditions for positive recurrence

Let Z = {Z(t), t ≥ 0} be a semimartingale reflecting Brownian motion that lives in the three-dimensional non-negative orthant. A 2002 paper by El Kharroubi, Ben Tahar and Yaacoubi gave sufficient conditions for positive recurrence of Z. Recently Bramson, Dai and Harrison have shown that those conditions are also necessary for positive recurrence. In this paper we provide an alternative proof of sufficiency, the salient feature of which is its use of a linear Lyapunov function.