Green’s Function For Dipolar Schramm-Loewner Evolution And Related Martingales

Stochastic Loewner Evolution or Schramm-Loewner Evolution(SLE),introduced by Schramm,is a one-parameter family of two-dimensional random growth process,which pro-vides a powerful mathematical tool for obtaining rigorous results about a various of discrete lattice models from statistical mechanics.The main work of this thesis is as follows:First,Green’s function of dipolar SLE is investigated.Fixed 0≤κ<8,Green’s function of dipolar SLE in the simple connected domain D is first defined.Next,based on dipolar Loewner differential equation,a partial differential equation and local martingales associated with Green’s function are constructed.Finally,the existence of Green’s function for dipolar SLE in the strip domain S_π={z∈C:0<(?)(z)<π}from 0 to the upper boundary R_π={z:(?)(z)=π}is proved and the expression of the Green’s function is given by terms of the expectation associated with dipolar SLE.Secondly,the associated martingales for dipolar SLE with forced points is discussed.Based on the system of stochastic differential equations satisfied by dipolar SLE_κwith the forced points and Girsanov’s theorem,local martingales associated with dipolar SLE_κare constructed,and dipolar SLE_κwith the forced points are obtained under the new measure weighted by the martingales.Moreover,two examples of martingales associated with the estimates of small probability events are given.