A POSITIVE RECURRENT REFLECTING BROWNIAN MOTION WITH DIVERGENT FLUID PATH

• 成果类型: Article
• 署名作者: Bramson, Maury
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP713
• 发表日期: 2011
• 页码: 951-986
• 关键词: stability; orthant

摘要

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 x d covariance matrix Sigma, and a d x 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 [Ann. Probab. 22 (1994) 680-702] 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, Ben Tahar and Yaacoubi [Stochastics Stochastics Rep. 68 (2000) 229-253, Math. Methods Oper. Res. 56 (2002) 243-258] gave sufficient conditions on (theta, Sigma, R) for positive recurrence for d = 3; Bramson, Dai and Harrison [Ann. Appl. Probab. 20 (2009) 753-783] 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.