We prove that a sufficient condition for a semimartingale reflecting Brownian motion in an orthant (SRBM) to be positive recurrent is that all solutions of an associated deterministic Skorokhod problem are attracted to the origin. To prove this result, we construct a Lyapunov function for the SRBM.
"Lyapunov Functions for Semimartingale Reflecting Brownian Motions." Ann. Probab. 22 (2) 680 - 702, April, 1994. https://doi.org/10.1214/aop/1176988725