| Stochastic Loewner evolution(SLE_κfor short)is a family of random curves with one-parameterκ>0,introduced by Schramm in 2000,which can be obtained by solving Loewner√differential equation when the driving function is a one-dimensional Brownian timesκ.SLE_κis intimately connected with scaling limits of a number of two-dimensional discrete models from statistical mechanics.Our main work in this paper is as follows.First,we investigate the derivative estimation of dipolar SLE_κ.based on that the coordinate transfor-mation between dipolar SLE_κand chordal one,from the reverse-time Loewner equation and the derivative estimation of chordal SLE_κwe derive a derivative estimation for dipolar SLE_κ.Secondly,the planar Brownian loop measure is studied.Using the properties of inverse radi-al SLE_κand probability measure,the Brownian loop measures in planar multiply-connected regions are discussed,and some related properties are obtained. |
