The Annals of Probability

Reflected Brownian Motion on an Orthant

J. Michael Harrison and Martin I. Reiman

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We consider a $K$-dimensional diffusion process $Z$ whose state space is the nonnegative orthant. On the interior of the orthant, $Z$ behaves like a $K$-dimensional Brownian motion with arbitrary covariance matrix and drift vector. At each of the (`K-1) dimensional hyperplanes that form the boundary of the orthant, $Z$ reflects instantaneously in a direction that is constant over that hyperplane. There is no extant theory of multidimensional diffusion that applies to this process, because the boundary of its state space is not smooth. We adopt an approach that requires a restriction on the directions of reflection, but Reiman has shown that this restriction is met by all diffusions $Z$ arising as heavy traffic limits in open $K$-station queuing networks. Our process $Z$ is defined as a path-to-path mapping of $K$-dimensional Brownian motion. From this construction it follows that $Z$ is a continuous Markov process and a semimartingale. Using the latter property, we obtain a change of variable formula from which one can develop a complete analytical theory for the process $Z$.

Article information

Ann. Probab. Volume 9, Number 2 (1981), 302-308.

First available in Project Euclid: 19 April 2007

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Primary: 60J50: Boundary theory
Secondary: 60J55: Local time and additive functionals 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Diffusion processes Markov processes reflecting barriers Brownian motion semimartingale Ito's formula queuing networks heavy traffic


Harrison, J. Michael; Reiman, Martin I. Reflected Brownian Motion on an Orthant. Ann. Probab. 9 (1981), no. 2, 302--308. doi:10.1214/aop/1176994471.

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