Brownian Motion on Compact Manifolds: Cover Time and Late Points

Let M be a smooth, compact, connected Riemannian manifold of dimension d‚ 3 and without boundary. Denote byT (x;†) the hitting time of the ball of radius † centered at x by Brownian motion on M. Then, C†(M) = supx2MT (x;†) is the time it takes Brownian motion to come within r of all points in M. We prove thatC†(M)=†2¡djlog†j! ∞dV (M) almost surely as †! 0, where V (M) is the Riemannian volume of M. We also obtain the \multi-fractal spectrum" f(fi) for \late points", i.e., the dimension of the set of fi-late points x in M for which limsup†!0T (x;†)=(†2¡djlog†j) = fi > 0.