On the Law of Iterated Logarithms for Brownian Motion on Compact Manifolds

Let M be a compact and smooth Riemannian manifold. Take a Brownian motion on the manifold. We take a family of measures defined by the Law of Iterated Logarithms for Brownian Motions on Compact Manifolds, as in Brosamler (1983). We study all of the accumulation points of this family of measures in the mild sense. We show for any subsequence of times, if we have mild convergence, then the limiting measure is absolutely continuous with respect to the volume measure on the manifold. We give a characterization of the limiting measure.