Dipolar Loewner Differential Equation And Its Related Questions

Stochastic Loewner evolution(SLE)is a one-parameter family of random fractal curves,introduced by Schramm in order to describe the scaling limit of loop erased random walk,which can be constructed by solving the Loewner differential equation when the driving term is a time-change of Brownian motion.SLE is a powerful tool to describe the scaling limits of discrete models from statistical physics.The main work of this thesis is as follows: First,the trace of dipolar Loewner equation and Lipschitz graph are investigated.it is proved that if W is a H �older-1/2 function with norm less than 4,then dipolar Loewner differential equation with W being the drive function generates a simple curve;a sufficient condition for driving functions of dipolar Loewner differential equation to generate curves which are graphs of Lipschitz functions is given.Secondly,the criterion for H �oder-1/2 function and the solution of second order differential equation are discussed.A criterion that driving functions of dipolar Loewner differential equation are H �oder-1/2 functions is given.The solution to a family of second order elliptic differential equation is constructed by using the inverse of dipolar SLE process.