Consider the first exit time and position from small geodesic balls for Brownian motion on Riemannian manifolds. We establish a smooth Besselization technique and calculate the asymptotic expansion for the joint distributions by a purely probabilistic approach.
"Wiener functionals associated with joint distributions of exit time and position from small geodesic balls." Ann. Probab. 24 (2) 825 - 837, April 1996. https://doi.org/10.1214/aop/1039639363