A natural parametrization for the Schramm–Loewner evolution

The Schramm-Loewner evolution (SLEκ) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When κ < 8, an instance of SLEκ 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 SLEκ that transforms like a d-dimensional volume measure under conformal maps. We prove that this parametrization is non-trivial (i.e., the curve is not entirely traversed in zero time) for κ < 4(7− √ 33) = 5.021 · · · .