Comparison theorems for Brownian motions on Riemannian manifolds and their applications

A global lower estimate for the transition probability of the Brownian motion on a complete Riemannian manifold is given in the form of a comparison with the transition probability of the Brownian motion on a model. This estimate is applied to obtain the laws of the iterated logarithms in the large for Brownian motion on certain Riemannian manifolds.