Eigenvalue expansions for Brownian motion with an application to occupation times. (English) Zbl 0891.60079

Summary: Let \(B\) be a Borel subset of \(\mathbb{R}^d\) 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 \(\lim_{u\to 0}\log P^0(A_1<u)/\log u=1/\xi\) 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.


60J65 Brownian motion
60J35 Transition functions, generators and resolvents
60J45 Probabilistic potential theory
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