Reversibility of Some Chordal SLE$(\kappa;\rho)$ Traces

TL;DR

This paper proves the reversibility of certain chordal SLE$(; ho)$ traces for specific parameters, extending understanding of their symmetry properties and providing equations for their reversal starting points.

Contribution

It establishes the reversibility of chordal SLE$(; ho)$ traces for  in (0,4) and  ho  (-4)/2, and derives equations for their reversal from arbitrary positive starting points.

Findings

Reversibility holds for  in (0,4) with  ho  (-4)/2.

Derived explicit equations for the reversal of SLE$(; ho)$ traces.

Extended the understanding of symmetry properties of SLE traces.

Abstract

We prove that, for $\kappa\in(0,4)$ and $\rho\ge (\kappa-4)/2$, the chordal SLE$(\kappa;\rho)$ trace started from $(0;0^+)$ or $(0;0^-)$ satisfies the reversibility property. And we obtain the equation for the reversal of the chordal SLE$(\kappa;\rho)$ trace started from $(0;b_0)$, where $b_0>0$.