General slit L\"owner chains

TL;DR

This paper extends L"owner theory to define new slit L"owner chains in the unit disk, enabling the construction of novel SLE-type processes with conformal invariance and domain Markov property, beyond classical examples.

Contribution

It introduces a generalized framework for slit L"owner chains that allows creating new SLE-type processes with key invariance properties.

Findings

New SLE-type processes with conformal invariance

Construction of processes with domain Markov property

Local behavior similar to classical SLE

Abstract

We use general L\"owner theory to define general slit L\"owner chains in the unit disk, which in the stochastic case lead to slit holomorphic stochastic flows. Radial, chordal and dipolar SLE are classical examples of such flows. Our approach, however, allows to construct new processes of SLE type that possess conformal invariance and the domain Markov property.The local behavior of these processes is similar to that of the classical SLE.