## Electronic Journal of Probability

### Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times

#### 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 $\lim_{u\to 0} \log P^0(A_1 < u) /\log u = 1/\xi$ 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.

#### Article information

Source
Electron. J. Probab., Volume 1 (1996), paper no. 3, 19 pp.

Dates
Accepted: 31 January 1996
First available in Project Euclid: 25 January 2016

https://projecteuclid.org/euclid.ejp/1453756466

Digital Object Identifier
doi:10.1214/EJP.v1-3

Mathematical Reviews number (MathSciNet)
MR1386295

Zentralblatt MATH identifier
0891.60079

Rights

#### Citation

Bass, Richard; Burdzy, Krzysztof. Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times. Electron. J. Probab. 1 (1996), paper no. 3, 19 pp. doi:10.1214/EJP.v1-3. https://projecteuclid.org/euclid.ejp/1453756466

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