SLE$_κ(ρ)$ bubble measures

Abstract

For $κ>0$ and $ρ>-2$, we construct a $σ$-finite measure, called a rooted SLE$_κ(ρ)$ bubble measure, on the space of curves in the upper half plane $\mathbb H$ started and ended at the same boundary point, which satisfies some SLE$_κ(ρ)$-related domain Markov property, and is the weak limit of SLE$_κ(ρ)$ curves in $\mathbb H$ with the two endpoints both tending to the root. For $κ\in(0,8)$ and $ρ\in ((-2)\vee(\fracκ2-4),\fracκ2-2)$, we derive decomposition theorems for the rooted SLE$_κ(ρ)$ bubble with respect to the Minkowski content measure of the intersection of the rooted SLE$_κ(ρ)$ bubble with $\mathbb R$, and construct unrooted SLE$_κ(ρ)$ bubble measures.