SLE${}_{\kappa }(\rho )$ bubble measures

Abstract

For $\kappa >0$ and $\rho >-2$, we construct a $\sigma$-finite measure, called a rooted SLE${}_{\kappa }(\rho )$ 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${}_{\kappa }(\rho )$-related domain Markov property, and is the weak limit of SLE${}_{\kappa }(\rho )$ curves in $\mathbb{H}$ with the two endpoints both tending to the root. For $\kappa \in (0,8)$ and $\rho \in ((-2)\vee (\frac{\kappa }{2}-4),\frac{\kappa }{2}-2)$, we derive decomposition theorems for the rooted SLE${}_{\kappa }(\rho )$ bubble with respect to the Minkowski content measure of the intersection of the rooted SLE${}_{\kappa }(\rho )$ bubble with $\mathbb{R}$, and construct unrooted SLE${}_{\kappa }(\rho )$ bubble measures.

51 pages. Added one figure. Added a list of notations at the end of Introduction. Rearranged some material in Sections 2 and 3. Added some explanation about the construction of SLE loop measures in Remark 3.9