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.
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