RADIAL PART OF BROWNIAN-MOTION ON A RIEMANNIAN MANIFOLD

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-epsilon be the firs t time t when rho(t) = epsilon. We show that E[rho(t boolean AND T)(2) ] = nt - (1/6)S(x)t(2) + o(t(2)), as t down arrow 0, where S(x) is the scalar curvature. The same formula holds for E[rho(t)(2)] under some boundedness condition on M.