Probability Seminar: Jim Dai (Georgia Institute of Technology)

On the Positive Recurrence of Semimartingale Reflecting Brownian Motions in Three Dimensions

Let Z be an n-dimensional Brownian motion confined to the non-negative orthant by oblique reflection at the boundary. Such processes arise in applied probability as diffusion approximations for multi-station stochastic processing networks. For dimension n = 2, a simple condition is known to be necessary and sufficient for positive recurrence of Z. The obvious analog of that condition is necessary but not sufficient in three and higher dimensions, where fundamentally new phenomena arise.

Building on prior work by Bernard and El Kharroubi (1991) and El Kharroubi et al. (2000, 2002), we provide necessary and sufficient conditions for positive recurrence in dimension n = 3. In this context we find that the fluid-stability criterion of Dupuis and Williams (1994) is not only necessary for positive recurrence but also sufficient; that is, in three dimensions Z is positive recurrent if and only if every path of the associated fluid model is attracted to the origin. I will also discuss recent development for problems in four and higher dimensions.  Joint work with Maury Bramson and Michael Harrison.

Refreshments will be served at 3:45pm in the 3rd floor lobby of Hanes Hall