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

Abstract

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