 Brownian loop soup

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]