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

Abstract

Schramm-Loewner evolution ( SLE κ ) 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.