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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.

Restrict $\mu$ first to loops surrounding $0$. This satisfies the pointed restriction property, so for one constant $C$,A single rooted pushforward cannot equal the whole ${\mu }_D$: it only sees loops surrounding its root. The rest is recovered by moving that root.

For $z\in D$, choose a disc automorphism ${\psi }_z$ with ${\psi }_z(0)=z$. Conformal restriction yieldswith the same constant. Let $(z_j)$ be a countable dense set in $D$. Every simple loop contained in $D$ surrounds one of these points. Make this cover disjoint by $P_j=E_{z_j}\setminus {\bigcup }_{i<j}{E}_{z_i}$. For a measurable event $A$ of loops in $D$,This explicitly relates the entire loop measure to the single-root measure without overcounting a loop once for every point in its interior. It also proves that knowledge of $\rho$, up to normalization, determines ${\mu }_D$. Nontriviality gives $C>0$: otherwise all countably many pointed patches would vanish. Using the usual unrooted Brownian loop measure, the same construction is commonly expressed as its outer-boundary pushforward, up to a constant.

The restriction and normalization make $\mu$ the unique conformal-restriction loop measure surrounding the origin, up to the fixed multiplicative normalization. By the classification of planar conformal restriction measures, it is the image of the rooted Brownian loop measure at the origin under the map taking a Brownian loop to its outer boundary. It is scale invariant and sigma-finite, with finite mass after imposing lower and upper diameter cutoffs.

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.