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 $\epsilon$-ball $B = \{ x : |x - b| < \epsilon\}$), via a function $h_M$ such that $d(A,B) = h_M(T_{AB}))$, where $T_{AB}$ is the average commute time from $A$ to $B$.

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