Basic properties of the natural parametrization for the Schramm-Loewner evolution

A number of measures on paths or clusters on two-dimensional lattices arising from critical statistical mechanical models are believed to exhibit some kind of conformal invariance in the scaling limit. Schramm introduced a one-parameter family of such processes, now called the (chordal) Schramm-Loewner evolution with parameter κ (SLEκ), and showed that these give the only possible limits for conformally invariant processes in simply connected domains satisfying a certain “domain Markov property”. He defined the process as a measure on curves from 0 to ∞ in H, and then used conformal invariance to define the process in other simply connected domains. The definition of the process in H uses parametrization by half-plane capacity (see Section 2.1 for definitions). Suppose γ : (0, t] → H is a (non-crossing) curve parametrized so that hcap(γ(0, t]) = at for some constant a > 0. We write γt for the set of points γ(0, t]. Let Ht denote the unbounded component of H \ γt and gt : Ht → H the unique conformal transformation with gt(z)− z = o(1) as z →∞. Then the following holds. • For z ∈ H, the map t 7→ gt(z) is a smooth flow and satisfies the Loewner differential equation ∂tgt(z) = a gt(z)− Ut , g0(z) = z,