Hitting Distributions of Small Geodesic Spheres

Abstract

Let $M$ be an $n$-dimensional Riemannian manifold, $m \in M$ and $T$ be the hitting time of an $r$-sphere around $m$ by Brownian motion $X_t$. We have, for any smooth function $g$ on the unit sphere $S$, under normal coordinates, $E^m \lbrack g(X_T/r) \rbrack = Ig + r^2I(\nu_2g) + r^3 I(\nu_3g) + O(r^4)$ and $E^m \lbrack Tg(X_T/r) \rbrack = E^m \lbrack T \rbrack E^m \lbrack g(X_T/r) \rbrack + r^5c \sum_i \partial_i sI(z_i g) + O(r^6)$, where $I$ is the uniform probability distribution on $S, \nu_2$ and $\nu_3$ are smooth functions on $S$ whose expressions involve scalar curvature, Ricci curvature and their derivatives at $m, c$ is a constant and $s$ is the scalar curvature. $\nu_2 = 0$ if and only if either $n = 2$ or $M$ is an Einstein manifold.