Optimal Hölder continuity and dimension properties for SLE with Minkowski content parametrization

We make use of the fact that a two-sided whole-plane Schramm–Loewner evolution (SLE κ ) curve γ for κ ∈ ( 0 , 8 ) from ∞ to ∞ through 0 may be parametrized by its d -dimensional Minkowski content, where d = 1 + κ 8 , and become a self-similar process of index 1 d with stationary increments. We prove that such γ is locally α -Hölder continuous for any α < 1 d . In the case κ ∈ ( 0 , 4 ] , we show that γ is not locally 1 d -Hölder continuous. We also prove that, for any deterministic closed set A ⊂ R , the Hausdorff dimension of γ ( A ) almost surely equals d times the Hausdorff dimension of A .