Uniform Dimension Theorem for SLE in Natural Time

Abstract

For $0<κ<8$, we prove a uniform dimension transformation theorem for SLE in natural time: on a single probability-one event, Hausdorff and packing dimensions of every time set are multiplied by $d=1+κ/8$, and the same holds for lower and upper box dimensions of every bounded time set. The result applies to self-similar SLE with stationary increments, chordal SLE in the half-plane, and chordal SLE in bounded piecewise $C^1$ Jordan domains. In the last case, the box-dimension conclusion holds under a positive-angle endpoint hypothesis. We also establish boundary-inclusive local Hölder continuity of every order below $1/d$ for chordal SLE in natural time and identify the exact local dimension $d$ of the natural occupation measure on compact time intervals.