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- ItemOn the regularity of SLE trace(Cambridge : Cambridge Univ. Press, 2017) Friz, Peter K.; Tran, HuyWe revisit regularity of SLE trace, for all κ≠8, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia–Rodemich–Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index min(1+κ/8,2), improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.
- ItemRegularity of SLE in (t,κ) and refined GRR estimates(Berlin ; Heidelberg ; New York, NY : Springer, 2021) Friz, Peter K.; Tran, Huy; Yuan, YizhengSchramm-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.