Lyapunov functions for semimartingale reflecting Brownian motions
Consider a nonnegative orthant \(S\) in \(R^ d\) in which a Brownian motion with constant drift \(r^ 0\) and covariance \(\Delta\) evolves. When it hits the \(i\)-th face of \(S\), it is reflected with direction \(r^ i\). Such a ``semimartingale reflecting Brownian motion'' (SRBM) associated with \((S,r^ 0, r^ 1, \dots, r^ d, \Delta )\) models certain problems in queueing networks under heavy traffic. The authors show that a sufficient condition for the SRBM to be positive recurrent is that all solutions of a related deterministic Skorokhod problem are attracted to the origin, i.e. the paths stay arbitrarily near zero after a time \(T\). The method of proving ergodicity uses the construction of a smooth Lyapunov function for the SRBM.
semimartingale reflecting Brownian motion
scientific article; zbMATH DE number 681378