The Stochastic Couplings Of SLE_κ Traces On The Strip Domain And Disk

The stochastic coupling technique is a powerful tool in studying reversibility of stochastic Loewner evolution (SLEκ). The main work of this paper is as follows:first, the stochastic coupling of chordal SLEκ(κ ∈ (0,4]) in the stirp region Sπ is discussed. Using a Feynman-Kac representation, we prove that there exists a coupling of two strip SLEκ traces in Sπ, one is from a to b, the other is from b to a, such that the two curves visit the same set of points. Secondly, the stochastic coupling of radial SLEκ(κ ∈ (0,4]) in the unit disk D is investigated. Based on a representation of solution to certain PDE, we construct a coupling of two radial SLEκ traces in D:γ1 and γ2, which may commute in the sense that, conditioned on one curve up to a finite time T, the other curve is radial SLEκ trace in the remaining region, and its marked point is the tip point of the first curve at T.