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 $\psi$-variation for $\psi (x)=x^d(\log 1/x{)}^{-d-\epsilon }$ and Hölder-type modulus $\phi (t)=t^{\alpha }(\log 1/t{)}^{\beta }$ where $d$ and $\alpha$ are the optimal $p$-variation and Hölder exponents of SLE${}_{\kappa }$ 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 $\phi (t)=(\log 1/t{)}^{-1/4}(\log \log 1/t{)}^{2+\epsilon }$. Finally, for $\kappa \ge 8$, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely ${\sup }_t|{\hat{f}}_t^{\prime }(u+iv)|\lesssim v^{2\alpha -1}(\log 1/v{)}^{\beta }$ in case $\kappa >8$ and $v^{-1}(\log 1/v{)}^{-1/4}(\log \log 1/v{)}^{1+\epsilon }$ in case $\kappa =8$. Our results are obtained from analysing the forward Loewner differential equation (in contrast to the other mentioned works which analyse the backward equation).

results have been improved in v2, notations have been made slightly more consistent

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Cited by in corpus (1)

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