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) = xd(log 1/x)−d−ε and Hölder-type modulus ϕ(t) = tα(log 1/t)β where d and α are the optimal p-variation and Hölder exponents of SLEκ which have been previously identified by Viklund, Lawler (Duke Math. J. 159 (2011) 351–383) and Friz, Tran (Forum Math. Sigma 5 (2017) e19). For SLE8, 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+ε. Finally, for κ ≥ 8, we prove regularity estimates for the uniformising maps that hold uniformly in time, namely supt |f̂t́(u + iv)| ≾ v2α−1(log 1/v)β in case κ > 8 and v−1(log 1/v)−1/4(log log 1/v)1+ε 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).