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 $B = \{ x : |x - b| < \epsilon\}$).

Possible ways of defining such a (quasi)metric:

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

* Let $d(a,b)$ be a function of the probability of reaching $B$ before time $T$. (suggested by [Arthur B](http://mathoverflow.net/users/8737/arthur-b))


In either case, is $d$ a metric?  Is it very different from geodesic distance?