A natural parametrization for the Schramm–Loewner evolution

Abstract

The Schramm–Loewner evolution (SLE_κ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ < 8, an instance of SLE_κ is a random planar curve with almost sure Hausdorff dimension d = 1 + κ/8 < 2. This curve is conventionally parametrized by its half plane capacity, rather than by any measure of its d-dimensional volume.

For κ<8, we use a Doob–Meyer decomposition to construct the unique (under mild assumptions) Markovian parametrization of SLE_κ that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is nontrivial (i.e., the curve is not entirely traversed in zero time) for $\kappa< 4(7-\sqrt{33})=5.021\ldots$.