Local Analysis of Loewner Equation

TL;DR

This paper establishes new conditions on the driving function of the Loewner equation that guarantee the generated process is a simple curve, extending previous results and analyzing cases involving Hölder-1/2 Weierstrass functions.

Contribution

It introduces improved criteria for simple curve generation in Loewner processes and explores duality relations between real and imaginary parts of the equation.

Findings

New conditions on $mbda$ for simple curve generation

Extension of results to $mbda(t)=cW_b(t)$ with Hölder-1/2 Weierstrass functions

Duality relation between real and imaginary parts of Loewner equation

Abstract

Let $\lambda:[0,+\infty)\mapsto\mathbb{R}$ be the driving function of a chordal Loewner process. In this paper we find new conditions on $\lambda$ which imply that the process is generated by a simple curve. This result improves former one by Lind ,Marshall and Rhode, and it particular gives new results about the case $\lambda(t)=cW_b(t)$, $W_b$ being a H\"{o}lder-$1/2$ Weierstrass function. In the second part we find new conditions on $\lambda$ implying that the process is generated by a curve. The main tool here is a duality relation between the real part and the imaginary part of the Loewner equation.