R^n中薄域上布朗运动期望寿命的渐近性
Asymptotics for the expected lifetime of Brownian motion on thin domains in R^n
摘要：
我们导出了R^n中薄区域上布朗运动的期望寿命和扭转刚度的三项渐近展开式，以及期望寿命的最大值（和相应的最大化者）的两项渐近展开式.该方法类似于我们以前用来研究Dirichlet Laplacian特征值的方法，包括在一个方向上缩放域，并在缩放参数趋于零时导出相应的渐近展开。除了沿着标度方向的一维布朗运动所支配外，我们还看到扰动的对称性在膨胀中起作用。在本征值的情况下，这些扩展也可以用来近似的退出时间的域的缩放参数不一定接近零。
英文摘要：
We derive a three-term asymptotic expansion for the expected lifetime of Brownian motion and for the torsional rigidity on thin domains in R^n, and a two-term expansion for the maximum (and corresponding maximizer) of the expected lifetime. The approach is similar to that which we used previously to study the eigenvalues of the Dirichlet Laplacian and consists of scaling the domain in one direction and deriving the corresponding asymptotic expansions as the scaling parameter goes to zero. Apart from being dominated by the one-dimensional Brownian motion along the direction of the scaling, we also see that the symmetry of the perturbation plays a role in the expansion. As in the case of eigenvalues, these expansions may also be used to approximate the exit time for domains where the scaling parameter is not necessarilly close to zero.
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