Decomposition of Schramm-Loewner evolution along its curve

We show that, for $\kappa \in (0,8)$, the integral of the laws of two-sided radial SLE${}_{\kappa }$ curves through different interior points against a measure with SLE${}_{\kappa }$ Green function density is the law of a chordal SLE${}_{\kappa }$ curve, biased by the path’s natural length. We also show that, for $\kappa >0$, the integral of the laws of extended SLE${}_{\kappa }(-8)$ curves through different interior points against a measure with a closed formula density restricted in a bounded set is the law of a chordal SLE${}_{\kappa }$ curve, biased by the path’s capacity length restricted in that set. Another result is that, for $\kappa \in (4,8)$, if one integrates the laws of two-sided chordal SLE${}_{\kappa }$ curves through different force points on $\mathbb{R}$ against a measure with density on $\mathbb{R}$, then one also gets a law that is absolutely continuous w.r.t. that of a chordal SLE${}_{\kappa }$ curve. To obtain these results, we develop a framework to study stochastic processes with random lifetime, and improve the traditional Girsanov’s Theorem.