| dc.description.abstract | In this thesis, we study a series of measures defined on the space of curves in the complex plane (or the equivalence classes of the curves space), including excursion measures, Boundary bubble measures, and Loop measures. We study the Loewner chain in the strip {? = ? + ?? ∈ ℂ ∶ ? ∈ (0, ?)}. We show the conformal restriction property for the dipolar SLE?(?) with a force point 0+, and we obtain an expression for the Radon-Nikodym derivative of dipolar SLE? in a domain with respect to dipolar SLE? in a subdomain for ? ∈ (0, 8/3].
The outline of this thesis is as follows: In Chapter 1, we briefly introduce the development of SLE. In Chapter 2, we introduce some properties about conformal transformations. We also introduce Brownian measures and the boundary Poisson kernels. Some calculations of Brownian measures for the dipolar case will be given. In Chapter 3, we first introduce the (chordal) Loewner differential equation and give the proofs for the dipolar case, and we show some calculations for the Loewner chain mapped by conformal transformations, and then introduce SLE process. In Chapter 4, we give the proofs of our main results. | en_US |