The Further Research On Some Properties Of Schramm-Loewner Evolution

Schramm-Lowener Evolution was introduced by Oded Schramm to resolve many problems of statistical physics and stochastic process in 2000, and then another two famous experts of probability, Gregory F. Lawler and Wendelin Werner, joined in and established this new researching branch together. This field intimatedly connected with some branches of science and technology, refers to statistical physics, probability, stochastic process, stochastic analysis and so on. The theory of SLE is still developing, so there are large amounts of problems which need to be studied. Our dissertation systematically researches basic properties of SLE . It is divided into three chapters.The first chapter is an introduction that is constituted by two parts. We simply introduce the main work of the famous authors in this field and explain corresponding lemmas and definitions respectively.In chapter two, we give a systematic study of SLE . The goal of section 1 is to obtain estimates for E [ gt' (z)a], and to derive the continuity properties of f? t(0). Section 2 proves a general criterion for the existence of a continuous trace. And then the proof that the SLE ktrace is continuous for k≠8 is completed. In section 3, we investigate the topological behavior of SLE . The trace is proved to be transient when k≠8 in section 4.In the third chapter, we estimate for the dimension of the trace and the boundary of the hull .