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Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times

   
@article{EJP3,
	author = {Richard Bass and Krzysztof Burdzy},
	title = {Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {1},
	year = {1996},
	keywords = {Brownian motion, eigenfunction expansion, eigenvalues, arcsine law.},
	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.},
	pages = {no. 3, 1-19},
	issn = {1083-6489},
	doi = {10.1214/EJP.v1-3},    
        url = {http://ejp.ejpecp.org/article/view/3}}