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

Bass, Richard F.; Burdzy, Krzysztof

Electron. J. Probab. 1, No. 3, 1-19 (1996).

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

Keywords:

eigenvalue expansion; Brownian motion killed on exiting; Brownian motion; eigenvalues of the Laplacian