The Expansion of Mean Distance of Brownian Motion on Riemannian Manifold

We study the asymptotic expansion in small time of the mean distance of Brownian motion on Riemannian manifolds. We compute the first four terms of the asymptotic expansion of the mean distance by using the decomposition of Laplacian into homogeneous components. This expansion can be expressed in terms of the scalar valued curvature invariants of order 2, 4, 6.