We use geometric microlocal methods to compute an asymptotic expansion of mean first arrival time for Brownian particles on Riemannian manifolds. This approach provides a robust way to treat this problem, which has thus far been limited to very special geometries. This paper can be seen as the Riemannian 3-manifold version of the planar result of [1] and thus enable us to see the full effect of the local extrinsic boundary geometry on the mean arrival time of the Brownian particles. Our approach also connects this question to some of the recent progress on boundary rigidity and integral geometry [21] and [18].

中文翻译：

---

黎曼流形上布朗粒子的平均首次到达时间

我们使用几何微局部方法来计算布朗粒子在黎曼流形上的平均首次到达时间的渐近展开。这种方法提供了解决此问题的可靠方法，迄今为止，该问题仅限于非常特殊的几何形状。本文可以看作是[1]的平面结果的黎曼三流形版本，因此使我们能够看到局部外在边界几何形状对布朗粒子平均到达时间的完全影响。我们的方法还将这个问题与边界刚度和整体几何学的一些最新进展联系起来[21]和[18]。