Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity

Nina Holden, Yizheng Yuan

Published 2022-11-28Version 1

We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrization. More precisely, denoting by $d\in(0,2]$ the dimension of the curve, we show the following. 1. The optimal $\psi$-variation is $\psi(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in the sense that $\eta$ is of finite $\psi$-variation for this $\psi$ and not for any function decaying more slowly as $x \downarrow 0$. 2. The optimal modulus of continuity is $\omega(s) = c\,s^{1/d}(\log s^{-1})^{1-1/d}$, i.e. $|\eta(t)-\eta(s)| \le \omega(t-s)$ for this $\omega$, and not for any function decaying more slowly as $s \downarrow 0$. 3. $\limsup_{t\downarrow 0} |\eta(t)|\,(t^{1/d}(\log\log t^{-1})^{1-1/d})^{-1}$ is a deterministic constant in $(0,\infty)$.