The Equivalence Of SLEs And Their Related Martingales

Stochastic Loewner evolution (SLE for short) is a one-parameter family of random con-formal invariant fractal curves obtained by solving a Loewner differential equation with the driving term being a time-change of one-dimensional Brownian motion. The process is inti-mately connected with scaling limits of percolation clusters. In this thesis our main work is as follows: Firstly, the equivalence of SLEs is investigated. Using conformal transformations it is derived that the distribution of dipolar SLE is the same as the distribution of chordal SLE stopped at some a.s. positive stopping time, modulo a time re-parameterization. Furthermore,it is deduced that dipolar SLE(K,?) with force points has the same law as a time change of chordal SLE(K,?) with force points, stopped at some a.s. positive stopping time. Secondly,we discuss some properties of martingales related to SLE(K,?) with force points. The proper-ties of the derivative of conformal map associated with hull is first investigated, and the limit of derivative of the conformal map at zero is obtained. Next, we construct some martingales corresponding to SLE(K;?) with force points and discuss some related properties.