Electronic Journal of Probability

Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times

Richard Bass and Krzysztof Burdzy

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65]
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Brownian motion eigenfunction expansion eigenvalues arcsine law

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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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