Refined regularity of SLE

Abstract

We prove refined (variation and Hölder-type) regularity statements for the SLE trace (under capacity parametrisation). More precisely, we show that the trace has finite $ψ$-variation for $ψ(x) = x^d(\log 1/x)^{-d-\varepsilon}$ and Hölder-type modulus $φ(t) = t^α(\log 1/t)^β$ where $d$ and $α$ are the optimal $p$-variation and Hölder exponents of SLE$_κ$ which have been previously identified by Viklund, Lawler (2011) and Friz, Tran (2017). For SLE$_8$, we simplify a step in the proof by Kavvadias, Miller, and Schoug (2021), and get the modulus $φ(t) = (\log 1/t)^{-1/4}(\log\log 1/t)^{2+\varepsilon}$. Finally, for $κ\ge 8$, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely $\sup_t |\hat f_t'(u+iv)| \lesssim v^{2α-1}(\log 1/v)^β$ in case $κ>8$ and $v^{-1}(\log 1/v)^{-1/4}(\log\log 1/v)^{1+\varepsilon}$ in case $κ=8$. Our results are obtained from analysing the forward Loewner differential equation (in contrast to the other mentioned works which analyse the backward equation).