2022/11/08 by Yu Pu, Yu, Pu
Mathematics · #60J67 #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2211.04581
openalex publication_date 2022/11/08 · openalex created_date 2022/11/15 · openalex updated_date 2026/07/28
Chordal SLEκ(\underlineρ) 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 ∞, whose law is influenced by additional force points on \mathbb R. When there are force points away from the origin, the law of SLEκ(\underlineρ) is not reversible as the ordinary chordal SLEκ. Zhan (2019) give an explicit description of the law of the time reversal of SLEκ(\underlineρ) 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 κ∈(0,8), the law of the time reversal of non-boundary filling SLEκ(\underlineρ) process is absolutely continuous with respect to SLEκ(\underlineρ) for some \underlineρ determined by \underlineρ, with the Radon-Nikodym derivative being a product of conformal derivatives.