  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 \cite{ammari} 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 [23,20].
