Some Properties Of SLE And Backward SLE On Strip Domain

Stochastic Loewner evolution(SLE or SLE_?for short) is a family of conformal invariant random fractal curves with one-parameter ?, which can be described by solving a Loewner differential equation when the driving function is a time-change of Brownian motion. For the study of SLE_?, the canonical SLE_?'s on the upper plane and the unit disk are generalized to ones on a general region. In this paper, our main work is as follows: first, the properties of SLE hull in the strip region are discussed. Using the properties of strip SLE and Schwarz reflection principe, we derive the relation between R-symmetric conformal mappings and hulls in the strip region. The relationship between the set which consists of a pair of disjoint hulls and Loewner conformal mappings is obtained. It is derived that the lift of a R-symmetric conformal mapping is continuous in the space of hulls in the strip region, and that some related mappings are continuous in the corresponding space of hulls, too. This generalizes the related properties of SLE hull in the upper half-plane to the case of strip region. Second, the properties of backward SLE on strip region are investigated. The backward SLE, forward Loewner chain and backward Loewner one on strip region are defined by Loewner differential equation. The properties of the forward Loewner chain and backward Loewner chain on strip region are discussed. The relationship between forward Loewner hull and forward Loewner chain is obtained, so does backward Loewner hull and backward Loewner chain. The random conformal weldings involving in some simple curves are discussed.