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 Electronic Journal of Probability > Vol. 1 (1996) > Paper 3 open journal systems 


Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times

Richard F. Bass, University of Washington
Krzysztof Burdzy, University of Washington


Abstract
Let $B$ be a Borel subset of $R^d$ with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting $B$. Let $A_1$ be the time spent by Brownian motion in a closed cone with vertex $0$ until time one. We show that $lim_{uto 0} log P^0(A_1 < u) /log u = 1/xi$ where $xi$ is defined in terms of the first eigenvalue of the Laplacian in a compact domain. Eigenvalues of the Laplacian in open and closed sets are compared.


Full text: PDF

Pages: 1-19

Published on: January 31, 1996


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Electronic Journal of Probability. ISSN: 1083-6489