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Brownian Analogues of Burke's Theorem

O'Connell, Neil; Yor, Marc


Keyword(s): quasireversibility; queues; Brownian motion; geometric functionals

Abstract: We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of an M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke's theorem for the Brownian queue was stated and proved by Harrison and Williams (1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the two dimensional case. In all of the above, reversibility plays a key role. Notes: Marc Yor, Laboratoire de Probabilites, Universite Pierre et Marie Curie, 4 Place Jussieu F- 75252 Paris Cedex 05, France

20 Pages

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