The Annals of Probability

Recurrence Classification and Invariant Measure for Reflected Brownian Motion in a Wedge

R. J. Williams

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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 $p(r, \theta) = r^{-\alpha}\cos(\alpha\theta - \theta_1)$.

Article information

Ann. Probab., Volume 13, Number 3 (1985), 758-778.

First available in Project Euclid: 19 April 2007

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Zentralblatt MATH identifier


Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J60: Diffusion processes [See also 58J65] 60J25: Continuous-time Markov processes on general state spaces

Brownian motion oblique reflection two-dimensional wedge fine recurrence transience invariant measure


Williams, R. J. Recurrence Classification and Invariant Measure for Reflected Brownian Motion in a Wedge. Ann. Probab. 13 (1985), no. 3, 758--778. doi:10.1214/aop/1176992907.

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