Regularity of SLE in $(t,\kappa )$ and refined GRR estimates

Abstract

Schramm-Loewner evolution (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-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (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 (GRR) inequality, which is of independent interest.

References in corpus (1)

• Remarks on Loewner Chains Driven by Finite Variation Functions

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