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

We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm–Loewner evolution (SLE): variation regularity, modulus of continuity and law of the iterated logarithm. For the latter two, we consider the SLE with its natural parametrization. More precisely, denoting by d∈(0,2] the dimension of the curve, we show the following: 1. The optimal ψ-variation is ψ(x)=xd(loglogx−1)−(d−1) in the sense that η is a.s. of finite ψ-variation for this ψ and not for any function decaying more slowly as x↓0. 2. The optimal modulus of continuity is ω(s)=cs1/d(logs−1)1−1/d, that is, for some random c>0 we have |η(t)−η(s)|≤ω(t−s) a.s., while this does not hold for any function ω decaying faster as s↓0. 3. lim supt↓0|η(t)|(t1/d(loglogt−1)1−1/d)−1 is a.s. equal to a deterministic constant in (0,∞). We also show that the natural parametrization of SLE is given by the fine mesh limit of the ψ-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity.