Time-reversal of multiple-force-point chordal $\mathrm{SLE}_\kappa(\underline{\rho})$

TL;DR

This paper proves Zhan's conjecture on the time-reversal law of chordal SLE$_()$ with multiple force points, showing it is absolutely continuous with respect to another SLE law, using Imaginary Geometry techniques.

Contribution

It confirms the conjecture that the time-reversal law of SLE$_()$ with force points is explicitly describable and relates to a different SLE law via conformal derivatives.

Findings

Time-reversal law of SLE$_()$ with force points is absolutely continuous to another SLE law.

Radon-Nikodym derivative is a product of conformal derivatives.

The conjecture by Zhan (2019) is proven.

Abstract

Chordal SLE$_\kappa(\underline{\rho})$ is a natural variant of chordal SLE curve. It is a family of random non-crossing curves on the upper half plane from 0 to $\infty$, whose law is influenced by additional force points on $\mathbb R$. When there are force points away from the origin, the law of SLE$_\kappa(\underline{\rho})$ is not reversible as the ordinary chordal SLE$_\kappa$. Zhan (2019) give an explicit description of the law of the time reversal of SLE$_\kappa(\underline{\rho})$ when all force points lies on the same sides of the origin, and conjectured that a similar result holds in general. In this paper we prove his conjecture. In particular, based on Zhan's result, using the techniques from the Imaginary Geometry developed by Miller and Sheffield (2013), we show that when $\kappa\in(0,8)$, the law of the time reversal of non-boundary filling $\mathrm{SLE}_\kappa(\underline{\rho})$ process is absolutely continuous with respect to $\mathrm{SLE}_\kappa(\underline{\hat{\rho}})$ for some $\underline{\hat{\rho}}$ determined by $\underline{\rho}$, with the Radon-Nikodym derivative being a product of conformal derivatives.