Boundary Green's functions and Minkowski content measure of multi-force-point SLE$_κ(\underlineρ)$

We consider a transient chordal SLE$_κ(ρ_1,\dots,ρ_m)$ curve $η$ in $\mathbb{H}$ from $w$ to $\infty$ with force points $v_1> \cdots >v_m$ in $(-\infty,w^-]$, which intersects and is not boundary-filling on $(-\infty,v_m)$. The main result is that there is an atomless locally finite Borel measure $μ_η$ on $η\cap (-\infty,v_m]$ such that for any $v<v_m$, the $d$-dimensional Minkowski content of $η\cap [v,v_m]$ exists and equals $μ_η[v,v_m]$, where $d=\frac{(\sum ρ_j+4)(κ-4-2\sum ρ_j)}{2κ}$ is the Hausdorff dimension of $η\cap [v,v_m]$. In the case that all $ρ_j=0$, this measure agrees with the covariant measure derived in [Alberts-Sheffield, 2011] for chordal SLE$_κ$ up to a multiplicative constant. %Such measure, called Minkowski content measure, satisfies conformal covariance properties. We call such measure a Minkowski content measure, extend it to a class of subsets of ${\mathbb{R}}^n$, and prove that they satisfy conformal covariance. To construct the Minkowski content measure on $η\cap [v,v_m]$, we follow the standard approach to derive the existence and estimates of the one- and two-point boundary Green's functions of $η$ on $(-\infty,v_m)$, which are the limits of the rescaled probability that $η$ passes through small discs or open real intervals centered at points on $(-\infty,v_m)$.

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