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Abstract

In this thesis, we study the properties of multiple-paths Schramm-Loewner Evolution (SLE). One of the main objectives is to study this process in multiply-connected domains, which requires discussing single path SLE in such domains first. ,While for some applications it is appropriate to consider SLE as a probability measure, there are several cases where it is more natural to consider it as a non-probability measure. In this work, we take the second approach to study multiple-paths SLE. The Brownian loop measure is one of the main ingredients of this work and we use it to give the Radon-Nikodym derivatives between several versions of SLE.,First, we discuss multiple-paths SLE in simply connected domains. In particular, we give a definition using the Brownian loop measure and show that the partition function is smooth. These results are based on a joint work with Greg Lawler. ,Next, we recall SLE in multiply-connected domains defined in a work of Lawler. As before, the Brownian loop measure is used to define SLE by describing particular Radon-Nikodym derivatives. In addition, we give an argument comparing SLE in annuli and radial SLE. ,Then, we define multiple-paths SLE in multiply-connected domains and prove that its partition function is smooth using the Hormander's theorem. While the definition is similar to multiple-paths SLE in simply-connected domains, the proof of smoothness of the partition functions in simply-connected domains cannot be easily extended to multiply-connected domains. This is because in multiply-connected domains, the explicit form of the partition function for two SLE paths is unknown. ,Finally, we use two independent radial SLE curves to give a construction of two-sided SLE measure growing simultaneously from the marked points. We show that this measure is comparable to the distribution of two SLE paths in an annulus as the inner circle shrinks.

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