Brownian loop measure

Brownian loop measure is a conformally invariant sigma-finite measure on unrooted planar Brownian loops. Taking outer boundaries connects it to conformal-restriction measures on self-avoiding loops.

In the unit disc, the relevant pointed Brownian loop measure is rooted at $0$ and normalized by its conformal deficit: for simply connected $U$ containing $0$, its mass of loops not contained in $U$ is minus the logarithm of the conformal radius. Its outer-boundary pushforward gives the corresponding pointed simple-loop restriction measure. It is not the full unrooted loop measure or a probability law of a fixed-duration Brownian bridge.

A sigma-finite measure on simple closed curves has conformal restriction if restricting to loops in a simply connected domain and then applying a conformal map agrees with first transporting the domain and restricting there. The usual nontrivial measures are finite on bounded macroscopic cutoff windows and are determined up to normalization by their pointed conformal deficits.

For simple loops surrounding a marked point, containment in two simply connected domains is containment in their intersection component containing that point. Finite deficits therefore determine all finite intersections of avoidance events by inclusion-exclusion. Restrict to finite windows $H_r={\mathbf{L}}_D\setminus {\mathbf{L}}_{rD}$ and use the sigma-finite uniqueness theorem for measures; as $r\downarrow 0$ these windows exhaust the pointed loop space. Finiteness of the windows is essential to this argument.