Brownian loop measure (${\mu }^{\mathrm{loop}}$)

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.

A Brownian loop soup of intensity $\lambda >0$ in a planar domain $D$ is a Poisson point process of unrooted Brownian motion loops with intensity measure $\lambda {\mu }_D^{\mathrm{loop}}$, where ${\mu }_D^{\mathrm{loop}}$ is Brownian loop measure restricted to loops contained in $D$. Unrooted means that a cyclic change of starting time does not produce a different loop.

For every measurable family $A$ of loops with ${\mu }_D^{\mathrm{loop}}(A)<\infty$, the number of soup loops in $A$ has Poisson distribution with mean $\lambda {\mu }_D^{\mathrm{loop}}(A)$. Counts in disjoint families are independent. Thus the probability that no loop belongs to $A$ is $\exp (-\lambda {\mu }_D^{\mathrm{loop}}(A))$. Finite restrictions give concrete simulation windows even though the whole intensity is infinite.

The conformal invariance of Brownian loop measure, with the appropriate change of time along each loop, gives conformal invariance of Brownian loop soup. Restricting a soup to loops contained in a smaller domain gives the soup there with the same intensity. This couples random loop geometry to conformal restriction measure on simple loops.^[ref]

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.

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.