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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.

Let $f:D\to D^{\prime }$ be a conformal map that is a bijection between planar domains. Mapping every loop of a Brownian loop soup in $D$ by $f$ gives a Brownian loop soup of the same intensity in $D^{\prime }$, after changing the clock along each loop. For a Brownian parametrization $\ell (t)$, the new clock isThe statement concerns unrooted loops: choosing a different starting time on the same loop does not create another object. It follows from conformal invariance of Brownian loop measure and the image property of a Poisson point process.^[ref]

For the dilation $f(z)=az$, spatial lengths multiply by $|a|$ and loop durations by $|a{|}^2$, while the soup intensity parameter stays fixed. Thus the random geometry has no preferred length scale. Keeping the clock change explicit prevents confusing conformal invariance of the loop ensemble with preservation of each loop's original duration.

Gregory F. Lawler studies random walks, Brownian motion and Schramm–Loewner evolution, with connections to statistical physics and complex analysis. His University of Chicago research profile emphasizes paths such as loop-erased random walks and their relation to uniform spanning trees.^[ref]

Lawler and Wendelin Werner developed the Brownian loop soup, using a conformally invariant measure on planar Brownian loops to build a Poisson point process of loops.^[ref] His Notes on the Bessel Process by Gregory F. Lawler provide another route from paths to geometry: a Bessel process describes radial diffusion, with its dimension parameter controlling whether the origin is reached. The boundary behavior matters when extending the process beyond its first zero.^[ref]

Wendelin Werner studies random geometric objects arising in probability theory and statistical physics, especially planar paths and their behavior under conformal maps. His work connects Brownian motion and Schramm–Loewner evolution with the geometry of random curves. His personal page identifies him at the University of Cambridge.^[ref]

With Gregory F. Lawler, Werner constructed the Brownian loop soup from a measure on unrooted planar Brownian loops. Sampling loops by a Poisson point process turns an invariant measure into a random collection of paths. The construction provides a way to add Brownian loops to a simple curve and study conformally invariant random geometry.^[ref]