1,341 research outputs found
Positive recurrence of reflecting Brownian motion in three dimensions
Consider a semimartingale reflecting Brownian motion (SRBM) whose state
space is the -dimensional nonnegative orthant. The data for such a process
are a drift vector , a nonsingular covariance matrix
, and a reflection matrix that specifies the boundary
behavior of . We say that is positive recurrent, or stable, if the
expected time to hit an arbitrary open neighborhood of the origin is finite for
every starting state. In dimension , necessary and sufficient conditions
for stability are known, but fundamentally new phenomena arise in higher
dimensions. Building on prior work by El Kharroubi, Ben Tahar and Yaacoubi
[Stochastics Stochastics Rep. 68 (2000) 229--253, Math. Methods Oper. Res. 56
(2002) 243--258], we provide necessary and sufficient conditions for stability
of SRBMs in three dimensions; to verify or refute these conditions is a simple
computational task. As a byproduct, we find that the fluid-based criterion of
Dupuis and Williams [Ann. Probab. 22 (1994) 680--702] is not only sufficient
but also necessary for stability of SRBMs in three dimensions. That is, an SRBM
in three dimensions is positive recurrent if and only if every path of the
associated fluid model is attracted to the origin. The problem of recurrence
classification for SRBMs in four and higher dimensions remains open.Comment: Published in at http://dx.doi.org/10.1214/09-AAP631 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Tutte's invariant approach for Brownian motion reflected in the quadrant
We consider a Brownian motion with drift in the quarter plane with orthogonal
reflection on the axes. The Laplace transform of its stationary distribution
satisfies a functional equation, which is reminiscent from equations arising in
the enumeration of (discrete) quadrant walks. We develop a Tutte's invariant
approach to this continuous setting, and we obtain an explicit formula for the
Laplace transform in terms of generalized Chebyshev polynomials.Comment: 14 pages, 3 figure
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