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VOL. 14 | 1986 Some positive eigenfunctions for elliptic operators with oblique derivative boundary conditions and consequences for the stationary densities of reflected Brownian motions
Ruth J. Williams

Editor(s) Michael Cowling, Christopher Meaney, William Moran

Abstract

Positive eigenfunctions for elliptic operators with oblique derivative boundary conditions arise as the stationary densities of reflected Brownian motions, briefly RBM's. An RBM is a diffusion process that behaves like Brownian motion with a constant drift velocity ~ in the interior of a d-dimensional domain and is instantaneously reflected at the boundary in a direction specified by a non-tangential vector field v on the boundary. When the domain is bounded and smooth and the vector field is smooth, it is shown that the stationary density is of a simple exponential form for all ~ if and only if the vector field v satisfies a certain skew-symmetry condition. A formal analogue of this result for polyhedral domains, where v is constant on each face, will also be given. Consequences for the existence and uniqueness of an RBM with such non-smooth data will be drawn from this.

Details will appear elsewhere.

Information

Published: 1 January 1986
First available in Project Euclid: 18 November 2014

Rights: Copyright © 1986, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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