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

TL;DR

This paper constructs and analyzes a new class of SLE$_appa( ho)$ bubble measures, establishing their properties, limits, and decompositions, advancing understanding of conformal invariance and fractal geometry in complex analysis.

Contribution

It introduces a novel rooted SLE$_appa( ho)$ bubble measure, proves its existence as a limit of curves, and develops decomposition theorems for these measures.

Findings

Constructed a $\sigma$-finite measure for SLE$_appa( ho)$ bubbles.

Proved the measure's domain Markov property and limit behavior.

Derived decomposition theorems with respect to Minkowski content.

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.