H(?)lder Regularity Of SLE_κ And The Related Estimate

Stochastic Loewner Evolution(SLEK for short)is a random growth process based on Loewner differential equation with driving parameter a time-change of one-dimensional Brow-nian motion.This process is intimately connected with the scaling limits of a number of two-dimensional discrete models from statistical mechanics and with the outer boundary of Brownian motion.Our main works in this paper are as follows.First,we investigate the H(?)lder regularity of strip SLEK trace.Based on Girsanov transformation and Borel-Cantelli Lemma we derive the H(?)lder regularity of strip SLEK with the optimal exponent,which gen-eralizes the H(?)lder regularity of SLEK from the chordal to the cases of strip.Next,we discuss an estimate of solution to radial Loewner differential equation.The estimation of the solution of the radial Loewner differential equation for time-direction is derived by using Bieberbach theorem,which generalizes the corresponding result for the chordal Loewner equation to the radial setting.