Loewner evolution driven by a stochastic boundary point

TL;DR

This paper studies a stochastic Loewner evolution in the unit disk driven by a boundary point, revealing that despite differences from SLE, the solutions share similar invariance properties.

Contribution

It introduces a new stochastic Loewner evolution driven by a boundary point, expanding understanding of boundary-driven conformal evolutions.

Findings

Solutions exhibit invariance properties similar to SLE.

The evolution differs from traditional SLE but maintains key invariance features.

Provides a framework for boundary-driven stochastic conformal maps.

Abstract

We consider evolution in the unit disk in which the sample paths are represented by the trajectories of points evolving randomly under the generalized Loewner equation. The driving mechanism differs from the SLE evolution, but nevertheless solutions possess similar invariance properties.