Some Estimates For Dipolar Loewner Chains And H(?)lder Regularity Of Their Traces

Stochastic Loewner evolution(SLE_κfor short)is a one-parameter family of random con-formal invariant fractal curves,which can be described by solving the classical Loewner d-ifferential equation with the driving term being a time-change of one-dimensional Brownian motion.The main work of this thesis is as follows:First,some estimates for dipolar Loewner chains and their traces are investigated.Based on the inverse dipolar Loewner equation it is first estimated that dipolar Loewner chains whose driving terms are continuous and weakly H(?)lder-1/2 continuous,respectively.Next,using Koebe’s distortion and one-quarter theorem-s,combined with the relation between the capacity and diameter of hull in a strip domain,the modulus of continuity for the trace of dipolar Loewner chains when the driving terms are weakly H(?)lder-1/2 continuous functions is estimated.Finally,a sufficient condition that dipolar Loewner chains when the driving terms are continuous functions are generated by a continuous path is given.Secondly,the optimal H(?)lder exponent for dipolar SLE_κtrace is dis-cussed.A local martingale associated with reverse-time dipolar SLE_κis constructed by using It(?)’s formula.From multifractal behavior analysis and the martingale constructed above,the estimates of moments for reverse-time dipolar SLE_κare obtained.This implies the optimal H(?)lder exponent for dipolar SLE_κtrace.