A boundary property of semimartingale reflecting Brownian motions

Summary We consider a class of reflecting Brownian motions on the non-negative orthant inR^K. In the interior of the orthant, such a process behaves like Brownian motion with a constant covariance matrix and drift vector. At each of the (K-1)-dimensional faces that form the boundary of the orthant, the process reflects instantaneously in a direction that is constant over the face. We give a necessary condition for the process to have a certain semimartingale decomposition, and then show that the boundary processes appearing in this decomposition do not charge the set of times that the process is at the intersection of two or more faces. This boundary property plays an essential role in the derivation (performed in a separate work) of an analytical characterization of the stationary distributions of such semimartingale reflecting Brownian motions.Research performed in part while the second author was visiting the Institute for Mathematics and Its Applications with support provided by the National Science Foundation and the Air Force Office of Scientific Research. R.J. William's research was also supported in part by NSF Grant DMS 8319562.