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 $d(a,b) = g(\epsilon) h_M(T_{aB})$, where $T_{aB}$ is the average commute time from $a$ to $B$, and $g(\epsilon)$ is the normalization function.

Is $d$ a metric? Is it very different from geodesic distance?