On the regularity of SLE trace

TL;DR

This paper investigates the regularity properties of Schramm-Loewner Evolution (SLE) traces, establishing optimal Besov, variation, and Hölder regularity results for all  eq 8, thereby improving previous bounds and understanding of SLE path smoothness.

Contribution

It provides new regularity results for SLE traces, including Besov, variation, and Hölder regularity, for all  eq 8, using advanced embedding theorems and moment estimates.

Findings

Established Besov regularity of SLE traces.

Proved optimal variation regularity with index ( + /8, 2).

Achieved almost sure regularity results improving previous bounds.

Abstract

We revisit regularity of SLE trace, for all $\kappa \neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia--Rodemich--Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1 + \kappa / 8, 2) $, improving on previous works of Werness, and also (optimal) H\"older regularity \`a la Johansson Viklund and Lawler.