Boundary Green's functions and Minkowski content measure of multi-force-point SLE${}_{\kappa }(\underset{‾}{\rho })$

We consider a transient chordal SLE${}_{\kappa }({\rho }_1,\dots ,{\rho }_m)$ curve $\eta$ in $\mathbb{H}$ from $w$ to $\mathrm{\infty }$ with force points $ v_1> \cdots

v_m$in$(-\infty,w^-]$,whichintersectsandisnotboundary-fillingon$(-\infty,v_m)$. The main result is that there is an atomless locally finite Borel measure ${\mu }_{\eta }$ on $\eta \cap (-\mathrm{\infty },v_m]$ such that for any $v<v_m$, the $d$-dimensional Minkowski content of $\eta \cap [v,v_m]$ exists and equals ${\mu }_{\eta }[v,v_m]$, where $d=\frac{(\sum \rho_j+4)(\kappa-4-2\sum \rho_j)}{2\kappa}$istheHausdorffdimensionof$\eta\cap [v,v_m]$. In the case that all ${\rho }_j=0$, this measure agrees with the covariant measure derived in [Alberts-Sheffield, 2011] for chordal SLE${}_{\kappa }$ 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 $\eta \cap [v,v_m]$, we follow the standard approach to derive the existence and estimates of the one- and two-point boundary Green’s functions of $\eta$ on $(-\mathrm{\infty },v_m)$, which are the limits of the rescaled probability that $\eta$ passes through small discs or open real intervals centered at points on $(-\mathrm{\infty },v_m)$.