# (quasi)metric on Riemannian manifolds via Brownian Motion?

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?

## 1 Answer

I think this will not be a metric if the space has a dimension > 1, because the Brownian commute times grow proportionally to r^d. This means you'll violate the triangle inequality: for example, if A, C, B are in a line on the plane and d(A, C) = d(C, B) = 1 and our scaling factor is conveniently 1, you'd have d(A, B) = 4 > 2 = d(A, C) + d(C, B).

• Can we instead use a function of the average commute time?
– gusl
Commented May 27, 2016 at 17:46
• The d-th root would make it possible to satisfy the triangle inequality, and should technically be a metric, but wouldn't seem to make practical sense. Commented May 27, 2016 at 17:51
• A metric based on the probability of reaching an epsilon ball in time t < T perhaps. Commented May 27, 2016 at 22:51
• Thanks Aaron and Arthur! I have edited the question to incorporate your suggestions.
– gusl
Commented May 27, 2016 at 23:49