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.

25 pages. The paper is revised according to the referees' comments

References in corpus (5)

• Duality of Chordal SLE
• Convergence of radial loop-erased random walk in the natural parametrization
• Two-sided radial SLE and length-biased chordal SLE
• Scaling limit of the loop-erased random walk Green's function
• Backward SLE and the symmetry of the welding

Cited by in corpus (2)

• Optimal Hölder Continuity and Dimension Properties for SLE with Minkowski Content Parametrization
• Decomposition of backward SLE in the capacity parameterization