## 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.

### MSC:

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