We prove the existence and nontriviality of the d-dimensional 4 Minkowski content for the Schramm-Loewner evolution (SLEκ) with κ < 8 and d = 1 + κ 8. We show that this is a multiple of the natural parameterization.1. Introduction. A number of measures on paths or clusters on twodimensional lattices arising from critical statistical mechanical models are believed to exhibit some kind of conformal invariance in the scaling limit. Schramm [13] introduced a one-parameter family of such processes, now called the (chordal) Schramm-Loewner evolution with parameter κ (SLE κ ) 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 probability 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 the half-plane Loewner equation. Suppose γ : (0, t] → H is a curve with γ(0) = 0, and let γ t = γ(0, t]. Let H t denote the unbounded component of H \ γ t . We assume that γ is noncrossing in the sense that for all s < t, γ[s, ∞) ⊂ H s , and γ[s, t] ∩ H s is nonempty. Let g t : H t → H be the unique conformal transformation with g t (z) − z = o(1) as z → ∞. Then for every a > 0, there exists a reparameterization of the curve such that the following holds:
In this paper, we will show that the higher moments of the natural parametrization of SLE curves in any bounded domain in the upper half plane is finite. We prove this by estimating the probability that an SLE curve gets near n given points.
The Green's function for the chordal Schramm-Loewner evolution SLE κ for 0 < κ < 8, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.
For a chordal SLE κ (κ ∈ (0, 8)) curve in a domain D, the n-point Green's function valued at distinct points z 1 , . . . , z n ∈ D is defined to bewhere d = 1 + κ 8 is the Hausdorff dimension of SLE κ , provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-constant bounds for them.A Proof of Theorem 3.1 46
Lemmas on two-sided radial SLEFor z ∈ H, and r > 0, we use P r z to denote the conditional law P[·|τ z r < ∞], and use P * z to denote the law of a two-sided radial SLE κ curve through z. For z ∈ R \ {0}, we use P * z to denote the law of a two-sided chordal SLE κ curve through z. Let E r z and E * z denote the corresponding expectation. In any case, we have P * z -a.s., T z < ∞. See [15,16] for definitions and more details on these measures. For a random chordal Loewner curve γ, we use (F t ) to denote the filtration generated by γ.
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