Radial Part of Brownian Motion on a Riemannian Manifold

Abstract

Let $\rho_t$ be the radial part of a Brownian motion in an $n$-dimensional Riemannian manifold $M$ starting at $x$ and let $T = T_\varepsilon$ be the first time $t$ when $\rho_t = \varepsilon$. We show that $E\lbrack \rho^2_{t\wedge T} \rbrack = nt - (1/6)S(x)t^2 + \sigma(t^2)$, as $t \downarrow 0$, where $S(x)$ is the scalar curvature. The same formula holds for $E\lbrack\rho^2_t\rbrack$ under some boundedness condition on $M$.