Abstract
We review one method for estimating the modulus of continuity of a Schramm–Loewner evolution (SLE) curve in terms of the inverse Loewner map. Then we prove estimates about the distribution of the inverse Loewner map, which underpin the difficulty in bounding the modulus of continuity of SLE for \(\kappa =8\). The main idea in the proof of these estimates is applying the Girsanov theorem to reduce the problem to estimates about one-dimensional Brownian motion.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lawler G, Schramm O, Werner W (2004) Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann Probab 32:939–995
Lawler G (2005) Conformally invariant processes in the plane. American Mathematical Society, Providence, RI
Lawler G (2009) Multifractal analysis of the reverse flow for the Schramm-Loewner evolution. Fractal geometry and stochastics IV. Birkhäuser, Boston, pp 73–107
Lind J (2008) Hölder regularity of the SLE trace. Trans Am Math Soc 360:3557–3578
Marshall D, Rohde S (2005) The Loewner differential equation and slit mappings. J Am Math Soc 18:763–778
Rohde S, Schramm O (2005) Basic properties of SLE. Ann Math (2) 161:883–924
Schramm O (2000) Scaling limits of loop-erased random walks and uniform spanning trees. Israel J Math 118:221–288
Viklund FJ, Lawler G (2011) Optimal Hölder exponent for the SLE path. Duke Math J 159:353–383
Viklund FJ, Lawler G (2012) Almost sure multifractal spectrum for the tip of an SLE curve. Acta Math 209:265–322
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of G. Lawler is supported by NSF Grant DMS-0907143.
Rights and permissions
About this article
Cite this article
Alvisio, M., Lawler, G.F. Note on the existence and modulus of continuity of the \({\textit{SLE}}_8\) curve. Metrika 77, 5–22 (2014). https://doi.org/10.1007/s00184-013-0471-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-013-0471-7