Abstract
Schramm–Loewner evolution (hbox {SLE}_kappa ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by sqrt{kappa } times Brownian motion. This yields a (half-plane) valued random field gamma = gamma (t, kappa ; omega ). (Hölder) regularity of in gamma (cdot ,kappa ;omega ), a.k.a. SLE trace, has been considered by many authors, starting with Rohde and Schramm (Ann Math (2) 161(2):883–924, 2005). Subsequently, Johansson Viklund et al. (Probab Theory Relat Fields 159(3–4):413–433, 2014) showed a.s. Hölder continuity of this random field for kappa < 8(2-sqrt{3}). In this paper, we improve their result to joint Hölder continuity up to kappa < 8/3. Moreover, we show that the SLE_kappa trace gamma (cdot ,kappa ) (as a continuous path) is stochastically continuous in kappa at all kappa ne 8. Our proofs rely on a novel variation of the Garsia–Rodemich–Rumsey inequality, which is of independent interest.
Highlights
Schramm–Loewner evolution (SLE) is a random path connecting two boundary points of a domain
Observe that since consider the difference only in the first parameter of G, the constant C in the statement of Lemma 2.1 does not depend on the size of [tk−1, tk], as we explained in Remark 2.3
Finall√y, we show that for each κ, the path γ (·, κ) is the SLEκ trace generated by κ B
Summary
Schramm–Loewner evolution (SLE) is a random (non-self-crossing) path connecting two boundary points of a domain. When we try to apply one of these versions to SLE (as a two-parameter random field in (t, κ)), we wish to estimate moments of |γ (t, κ) − γ (s, κ )|, where we denote the SLEκ trace by γ (·, κ). With the methods of this paper, it would require a better moment estimate in the style of (1) with larger exponent on the right-hand side If such an estimate were to hold true with arbitrarily large exponent on the right-hand side (and any suitable exponent on the left-hand side), which is not clear to us, almost sure continuity of the random field in all (t, κ) with κ = 8 would follow
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