On the Mean First Arrival Time of Brownian Particles on Riemannian Manifolds

TL;DR

This paper develops a geometric microlocal method to asymptotically analyze the mean first arrival time of Brownian particles on Riemannian manifolds, extending previous special-case results to more general geometries.

Contribution

It introduces a robust geometric approach to compute mean first arrival times on Riemannian manifolds, linking boundary geometry with stochastic processes.

Findings

Provides an asymptotic expansion for mean first arrival time

Extends planar results to Riemannian 3-manifolds

Connects boundary geometry with stochastic analysis

Abstract

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].