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

Abstract

Schramm–Loewner evolution (${\text{SLE}}_{\kappa }$) 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 $\gamma (·,\kappa ;\omega$), 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 $\gamma (·,\kappa )$ (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.