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August, 1985 Recurrence Classification and Invariant Measure for Reflected Brownian Motion in a Wedge
R. J. Williams
Ann. Probab. 13(3): 758-778 (August, 1985). DOI: 10.1214/aop/1176992907


The object of study in this paper is reflected Brownian motion in a two-dimensional wedge with constant direction of reflection on each side of the wedge. The following questions are considered. Is the process recurrent? If it is recurrent, what is its invariant measure? Let $\xi$ be the angle of the wedge $(0 < \xi < 2\pi)$ and let $\theta_1$ and $\theta_2$ be the angles of reflection on the two sides of the wedge, measured from the inward normals towards the directions of reflection, with positive angles being toward the corner $(-\pi/2 < \theta_1, \theta_2 < \pi/2)$. Set $\alpha = (\theta_1 + \theta_2)/\xi$. Varadhan and Williams (1985) have shown that the process exists and is unique, in the sense that it solves a certain submartingale problem, when $\alpha < 2$. It is shown here that if $\alpha < 0$, the process is transient (to infinity). If $0 \leq \alpha < 2$, the process is shown to be (finely) recurrent and to have a unique (up to a scalar multiple) $\sigma$-finite invariant measure. It is further proved that the density for this invariant measure is given in polar coordinates by $p(r, \theta) = r^{-\alpha}\cos(\alpha\theta - \theta_1)$.


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R. J. Williams. "Recurrence Classification and Invariant Measure for Reflected Brownian Motion in a Wedge." Ann. Probab. 13 (3) 758 - 778, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0596.60078
MathSciNet: MR799421
Digital Object Identifier: 10.1214/aop/1176992907

Primary: 60J65
Secondary: 60J25 , 60J60

Keywords: Brownian motion , fine recurrence , invariant measure , Oblique reflection , transience , two-dimensional wedge

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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