Schramm–Loewner evolution (\(\hbox {SLE}_\kappa \)) 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 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 \(\gamma (\cdot ,\kappa )\) (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 inequality, which is of independent interest.

Schramm–Loewner 演化 ( \(\hbox {SLE}_\kappa \) ) 是通过 Loewner 演化与半平面容量参数化进行经典研究的，由\(\sqrt{\kappa }\)倍布朗运动驱动。这产生了一个（半平面）值随机场\(\gamma = \gamma (t, \kappa ; \omega )\)。(Hölder) in \(\gamma (\cdot ,\kappa ;\omega \) ) 的(Hölder) 正则性，又名 SLE 轨迹，已被许多作者考虑，从 Rohde 和 Schramm (Ann Math (2) 161(2) 开始： 883–924, 2005)。随后，Johansson Viklund 等人。(Probab Theory Relat Fields 159(3–4):413–433, 2014) 显示为\(\kappa < 8(2-\sqrt{3})\)这个随机场的 Hölder 连续性。在本文中，我们将他们的结果改进为联合 Hölder 连续性高达\(\kappa < 8/3\)。此外，我们表明 SLE \(_\kappa \)轨迹\(\gamma (\cdot ,\kappa )\)（作为连续路径）在\(\kappa \)中完全随机连续\(\kappa \ne 8\)。我们的证明依赖于 Garsia-Rodemich-Rumsey 不等式的一种新颖变体，它具有独立的意义。