Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to some property of Brownian Motion from $a$ to $b$ (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)

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