Given points A and B on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to the average commute time from A to B under Brownian Motion (or rather, to an $\epsilon$-ball around B), via a function $g_M$ such that d(A,B) = g(average_commute_time(A,B)).

After appropriate normalization, do we have a metric? Would this be very different from geodesic distance?