Stochastic Loewner Evolution Driven By Lévy Skew Stable Processes

A stochastic Loewner evolution(SLEκfor short) is a one-parameter family of random planar growth processes constructed by solving the Loewner equation when the driving function is a one-dimension Brownian motion. This process is intimately connected with scaling limits of percolation clusters. In this thesis we investigate the Loewner differential equation when the driving function is equal to the sum of a onedimension Brownian motion and a Lévy skew stable process. We derive that the derivatives estimates of conformal mappings for the backward flow of the Loewner equation.The pathwise increment estimates of the Lévy skew stable process are obtained. It is proved that the corresponding increasing hull is generated by a curve which is right continuous with left limit.