Brownian Motion And Dipolar Schramm Loewner Evolution

Stochastic Loewner evolution(SLE for short) is a family of random planar growth processes with one-parameter, which can be described by solving a Loewner differential equation with the driving term being a time-change of one-dimensional Brownian motion. This process is closely related to the scaling limits of percolation clusters and Brownian motion. In this thesis, our main work is as follows. First, the estimations of crossing probabilities for Brownian motions are discussed. Using the properties of Brownian motions, it is derived that the probability expressions of two-dimensional Brownian motion exiting first at prescribed boundary parts of simply connected region. The correctness of the obtained results is verified by numerical simulations; Secondly, the properties of dipolar SLE are investigated. The expressions of crossing probability for the dipolar SLE_?(? ? 4) in the strip domain S_?, unit disk domain D, half plane domain H and upper-half unit disk domain D_? are gived, respectively.The formulas of hitting probability density for dipolar SLE_?(? > 0) are derived; In addition,it is proved that dipolar SLE6 has local property. The restriction property of dipolar SLE8/3and other several related properties are discussed. Finally, the relationships between some two-dimensional statistical physics models and SLEs are investigated, and the ones between traces of chordal SLEs and some two-dimensional statistical physics models are simulated by the Monte Carlo method.