A support theorem for SLE curves

TL;DR

This paper proves that the support of SLE$_ppa$ curves can be characterized as the closure of Loewner curves driven by smooth functions, linking SLE support to simple curves in a rigorous way.

Contribution

It establishes a support theorem for SLE curves, connecting their support to smooth-driven Loewner curves, which was previously not well-understood.

Findings

Support of SLE$_ppa$ is the closure of smooth-driven Loewner curves.

Support includes the closure of simple curves starting at 0.

Provides a rigorous foundation for approximating SLE with smooth curves.

Abstract

For all $\kappa > 0$, we show that the support of SLE$_\kappa$ curves is the closure in the sup-norm of the set of Loewner curves driven by nice (e.g. smooth) functions. It follows that the support is the closure of the set of simple curves starting at $0$.