Decomposition of Schramm-Loewner evolution along its curve

Abstract: 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.