TITLE: Is there a probabilistic interpretation of McShane's identity?
QUESTION [6 upvotes]: Recall McShane's Identity.
Take a Brownian Motion on this punctured torus constrained to return to the original point. Now take the "curve" you get this way and try and contract it as much as possible. Maybe the probability you get of ending with a given simple closed geodesic of length $l$ is $2/(1+e^l)$ or something.
This would give a probabilistic proof of McShane's identity or something like this.
Can one construct this?

REPLY [3 votes]: Not that I know of, but there is a Brownian motion argument for the (philosophically related) Basmajian identity:
Calegari, Danny, Chimneys, Leopard spots and the identities of Basmajian and Bridgeman, Algebr. Geom. Topol. 10, No. 3, 1857-1863 (2010). ZBL1196.57010.
The Brownian motion thing seems to be not really essential, as shown in my preprint. But make of all this what you will.