Reversibility of chordal SLE with force points

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Let β(t)\beta(t), 0≤t<∞0\le t<\infty, be a chordal SLE(κ;ρ−,ρ+)(\kappa;\rho_-,\rho_+) trace started from (0;0−,0+)(0;0^-,0^+), where

κ∈(0,4),ρ−,ρ+≥κ/2−2.\kappa\in(0,4),\qquad \rho_-,\rho_+\ge \kappa/2-2.

Let W(z)=−1/zW(z)=-1/z.

Reversibility of chordal SLE with force points. After a time-change, the reversal of (W(β(t)))(W(\beta(t))) has the same distribution as a chordal SLE(κ;ρ−,ρ+)(\kappa;\rho_-,\rho_+) trace started from (0;0+,0−)(0;0^+,0^-).

This asserts that inversion in the unit semicircle, followed by time reversal, preserves the law of the force-point SLE trace while interchanging the two force points. The supplied text does not state whether the result is proved or remains open.

References

Primary source

Dapeng Zhan, “Duality of Chordal SLE, II”, arXiv:0803.2223 (2008).

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