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

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. ${\mathrm{limsup}}_{t\downarrow 0}|\eta (t)|(t^{1/d}(\log \log t^{-1}{)}^{1-1/d}{)}^{-1}$ 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.

paper is extended in v3

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