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 $lim_{uo 0} log P^0(A_1 < u) /log u = 1/xi$ 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.
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机译：令$ B $为具有有限体积的$ R ^ d $的Borel子集。我们给出了在退出$ B $时被杀死的布朗运动的跃迁密度的特征值展开。设$ A_1 $是布朗运动在顶点为$ 0 $的闭合圆锥中直到时间一所花费的时间。我们证明$ lim_ {u to 0} log P ^ 0（A_1 展开▼
