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Brownian motion in a wedge with oblique reflection. (English) Zbl 0579.60082

This work is concerned with the existence and uniqueness of a strong Markov process with continuous sample paths that has the following additional properties: (i) the state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like a Brownian motion in \({\mathbb{R}}^ 2\), (ii) the process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side, (iii) the amount of time the process spends at the corner of the wedge is zero.
Let \(\alpha =(\theta_ 1+\theta_ 2)/\xi\), \(\theta_ 1\) and \(\theta_ 2\) being the angles of reflection on the two sides of the wedge, \(\xi\) being the angle of the wedge. It is shown that there exists a unique continuous strong Markov process satisfying (i) and (ii) (but no such process satisfying (i)-(iii)) if \(\alpha\geq 2\). In this case the process reaches the corner of the wedge almost surely and remains there. If \(\alpha <2\) there exists a unique continuous strong Markov process satisfying (i)-(iii). It is shown that starting away from the corner, this process does not reach the corner of the wedge if \(\alpha\leq 0\), and does reach the corner if \(0<\alpha <2.\)
The general theory of diffusions does not apply to this situation because the boundary of the state space is not smooth and there is a discontinuity in the direction of the reflection at the corner. For some values of \(\alpha\), the process arises from diffusion approximations to storage systems and queuing networks.
Reviewer: M.Dozzi

MSC:

60J65 Brownian motion
60J60 Diffusion processes
60G44 Martingales with continuous parameter
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