An Introduction to the Loewner Equation and SLE

Abstract

This paper is meant to serve as an introduction and reference guide to the Loewner equation and Stochastic Loewner Evolution (SLE). We begin by discussing various forms of Loewner’s differential equation and some of the technical language of the surrounding mathematics. After proving the ability for the equation to generate slits in the upper-halfplane we investigate some properties and theorems that will be beneficial when we deal with SLE. Our attention is then turned to the realm of stochastic processes as we introduce Brownian motion and how it is used as a driving term to generate SLE. Since modeling SLE is (at the time of this publication) our primary endeavor, I have tried to use the mathematics developed in the previous sections to explain how this can be achieved both theoretically and in practice. Finally, we look at a specific example of a simple program that generates SLE for various values of κ. It is this program that serves as a base for modeling SLE via Mathematica. Although the more recent programs have made a number of developments, they all still use the same basic idea so it will be beneficial to understand how it works. This paper was targeted toward those who have an understanding of real and complex analysis. However, an introduction to probability would also be beneficial. Beyond this I have tried to make it as self-contained as possible and have included two appendices to supplement some of the more specific background information.

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Cite this paper

@inproceedings{Duncan2006AnIT, title={An Introduction to the Loewner Equation and SLE}, author={David Duncan}, year={2006} }