Eigenvalue Expansions for Brownian Motion with an Application to Occupation Times
نویسندگان
چکیده
Let B be a Borel subset of R with finite volume. We give an eigenvalue expansion for the transition densities of Brownian motion killed on exiting B. Let A1 be the time spent by Brownian motion in a closed cone with vertex 0 until time one. We show that limu→0 logP (A1 < u)/ log u = 1/ξ where ξ 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.
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