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

Abstract

Schramm-Loewner evolution ( SLEκ ) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by κ 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 SLE κ 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.