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

TL;DR

This paper establishes optimal bounds for the regularity measures of Schramm-Loewner evolution (SLE), including variation, modulus of continuity, and law of the iterated logarithm, with results valid for the natural parametrization.

Contribution

It provides the first sharp bounds for SLE regularity measures and connects the natural parametrization with the $ ext{psi}$-variation limit.

Findings

Optimal $ ext{psi}$-variation characterized by $x^d( ext{log} ext{log} x^{-1})^{-(d-1)}$

Modulus of continuity given by $s^{1/d}( ext{log} s^{-1})^{1-1/d}$

Law of the iterated logarithm limit is a positive finite constant

Abstract

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 parametrisation. 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 a.s. 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. for some random $c>0$ we have $|\eta(t)-\eta(s)| \le \omega(t-s)$ a.s., while this does not hold for any function $\omega$ decaying faster as $s \downarrow 0$. 3. $\limsup_{t\downarrow 0} |\eta(t)|\,\big(t^{1/d}(\log\log t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in $(0,\infty)$. We also show that the natural parametrisation of SLE is given by the fine mesh limit of the $\psi$-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity.