 Brownian loop measure

Brownian loop measure is a sigma-finite measure on unrooted planar Brownian motion loops. A loop has a duration $t>0$; changing its starting time cyclically represents the same unrooted loop. Let ${\mathbb{P}}_{z,t}^{\mathrm{br}}$ be the probability law of a planar Brownian bridge from $z$ to $z$ of duration $t$. For a nonnegative measurable function $F$ of an unrooted loop,Here $dA$ is planar Lebesgue measure. The weight combines the planar Brownian transition density $1/(2\pi t)$ with a further factor $1/t$ for unrooting. Restricting to loops contained in a domain gives its loop measure. A conformal map, with the conformal Brownian clock on each loop, transports this restricted measure to that of the image domain.^[ref]

A Poisson point process with intensity $\lambda {\mu }^{\mathrm{loop}}$ is a Brownian loop soup. This construction connects Brownian motion to conformal restriction measure on simple loops through loop boundaries. The measure is infinite, so drawing one loop requires a finite restriction and normalization.