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Regularity of SLE in (t,κ) and refined GRR estimates

Friz, Peter K.; Tran, Huy; Yuan, Yizheng

AG Stochastik und Finanzmathematik

Schramm–Loewner evolution (SLEκ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by √k times Brownian motion. This yields a (half-plane) valued random field γ=γ(t,κ;ω). (Hölder) regularity of in γ(⋅,κ;ω), 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 κ<8(2−√3). In this paper, we improve their result to joint Hölder continuity up to κ<8/3. Moreover, we show that the SLEκ trace γ(⋅,κ) (as a continuous path) is stochastically continuous in κ at all κ≠8. Our proofs rely on a novel variation of the Garsia–Rodemich–Rumsey inequality, which is of independent interest.