A Positive Recurrent Reflecting Brownian Motion with Divergent Fluid Path

TL;DR

This paper constructs examples of six-dimensional SRBMs that are positive recurrent despite having diverging fluid paths, challenging previous assumptions about the relationship between fluid path behavior and recurrence.

Contribution

It provides the first known examples in dimension six where SRBMs are positive recurrent with diverging fluid paths, showing the converse of a key recurrence criterion does not hold.

Findings

Positive recurrence can occur with diverging fluid paths in d=6 SRBMs.

The converse of the Dupuis-Williams criterion does not hold for dimensions six and higher.

Examples demonstrate the complexity of SRBM recurrence behavior in higher dimensions.

Abstract

Semimartingale reflecting Brownian motions (SRBMs) are diffusion processes with state space the d-dimensional nonnegative orthant, in the interior of which the processes evolve according to a Brownian motion, and that reflect against the boundary in a specified manner. The data for such a process are a drift vector {\theta}, a nonsingular d \times d covariance matrix {\Sigma}, and a d \times d reflection matrix R. A standard problem is to determine under what conditions the process is positive recurrent. Necessary and sufficient conditions for positive recurrence are easy to formulate for d = 2, but not for d > 2. Associated with the pair ({\theta}, R) are fluid paths, which are solutions of deterministic equations corresponding to the random equations of the SRBM. A standard result of Dupuis and Williams [6] states that when every fluid path associated with the SRBM is attracted to the origin, the SRBM is positive recurrent. Employing this result, El Kharroubi et al. [7, 8] gave sufficient conditions on ({\theta},{\Sigma},R) for positive recurrence for d = 3; Bramson et al. [2] showed that these conditions are, in fact, necessary. Relatively little is known about the recurrence behavior of SRBMs for d > 3. This pertains, in particular, to necessary conditions for positive recurrence. Here, we provide a family of examples, in d = 6, with {\theta} = (-1, -1, . >. ., -1)T, {\Sigma} = I and appropriate R, that are positive recurrent, but for which a linear fluid path diverges to infinity. These examples show in particular that, for d >= 6, the converse of the Dupuis-Williams result does not hold.