Diffusions with reflection on an orthant and associated initial-boundary value problems

Abstract

A class of diffusion processes with instantaneous reflection on the hyperplanes of an orthant is considered. In the case of constant drift and covariance coefficients, such diffusions arise as limits of properly normalized queue length processes in open queueing networks, under heavy traffic conditions. The directions of the reflection on each hyperplane determine the boundary data in an associated initial-boundary value problem with oblique derivatives for the corresponding Kolmogorov equation. This problem is studied in terms of single layer potentials with densities placed on the hyperplanes of the orthant, and is shown to be equivalent to solving a system of integral equations for those densities. The stochastic representation of the solution is also derived.