##
**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.

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

### Keywords:

submartingale problem; existence and uniqueness of a strong Markov process; two-dimensional wedge; storage systems and queuing networks
PDFBibTeX
XMLCite

\textit{S. R. S. Varadhan} and \textit{R. J. Williams}, Commun. Pure Appl. Math. 38, 405--443 (1985; Zbl 0579.60082)

Full Text:
DOI

### References:

[1] | and , Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series, 55, 1972. |

[2] | Anderson, Nagoya Math. J. 60 pp 189– (1976) · Zbl 0324.60063 |

[3] | On Reflected Brownian Motion in Two Dimensions, Ph. D. Dissertation, Department of Statistics, University of California at Berkeley, California, 1975. |

[4] | and , An Introduction to Stochastic Integration, Birkhäser, Boston, 1983. |

[5] | and , Probabilities and Potential, Vol. I, North-Holland, Amsterdam, 1978. |

[6] | Harrison, Annals Prob. 9 pp 302– (1981) |

[7] | Keller, SIAM J. Appl. Math. 41 pp 294– (1981) |

[8] | Approximation of Population Processes, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 1981. · Zbl 0465.60078 |

[9] | Lauwerier, I. Kon. Ned. Akad. v. Wet. Proc. A62 pp 475– (1959) |

[10] | II. Kon. Ned. Akad. v. Wet. Proc. A63 pp 355– (1960) |

[11] | Lindvall, J. Appl. Prob. 10 pp 109– (1973) |

[12] | Brownian motions in the half-Plane with singular inclined periodic boundary conditions, Topics in Probability Theory, ed. and , New York University, 1973, pp. 163–179. |

[13] | Prokhorov, Theor. Prob. and Applic. 1 pp 157– (1956) |

[14] | Reiman, Mathematics of Operations Research. · Zbl 0421.30015 |

[15] | Stroock, Comm. Pure Appl. Math. 24 pp 147– (1971) |

[16] | and , Multi-Dimensional Diffusion Processes, Springer-Verlag, New York, 1979. |

[17] | On the stochastic differential equation for a Brownian motion with oblique reflection on the half-Plane, Proc. of International Symposium on Stochastic Differential Equations, Kyoto, 1976, Ed., Wiley-Interscience, New York, 1978, pp. 427–436. |

[18] | Tsuchiya, J. Math. Soc. Japan 32 pp 233– (1980) |

[19] | Van Dantzig, Kon. Ned. Akad. v. Wet. Proc. A61 pp 384– (1958) |

[20] | Brownian motion in a wedge with oblique reflection at the boundary, Ph. D. Dissertation, Stanford University, 1983. |

[21] | de Zelicourt, Ann. Inst. Henri Poincaré 17 pp 351– (1981) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.