Decomposition of backward SLE in the capacity parameterization

Abstract

We prove that, for $\kappa \le 4$, backward chordal SLE${}_{\kappa }$ admits backward chordal SLE${}_{\kappa }(-4,-4)$ decomposition for the capacity parametrization. This means that, for any bounded measurable subset $U\subset Q_4:={\mathbb{R}}_{+}\times {\mathbb{R}}_{-}$, if we integrate the laws of extended backward chordal SLE${}_{\kappa }(-4,-4)$ with different pairs of force points $(x,y)$ against some suitable density function $G(x,y)$ restricted to $U$, then we get a measure, which is absolutely continuous with respect to the law of backward chordal SLE${}_{\kappa }$, and the Radon-Nikodym derivative is a constant depending on $\kappa$ times the capacity time that the generated welding curve $t\mapsto (d_t,c_t)$ spends in $U$, where $d_t>0>c_t$ are the pair of points that are swallowed by the process at time $t$. For the forward SLE curve, a similar analysis has been done for SLE in the natural parametrization ([1] $\kappa \le 4$, [10] $\kappa <8$), and for the capacity parametrization ([10] $\kappa <\infty$).

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