Regularity of Schramm-Loewner evolutions, annular crossings, and rough path theory

Brent Morehouse Werness (University of Washington)


When studying stochastic processes, it is often fruitful to understand several different notions of regularity.  One such notion is the optimal Hölder exponent obtainable under reparametrization.  In this paper, we show that chordal $\mathrm{SLE}_\kappa$ in the unit disk for $\kappa \le 4$ can be reparametrized to be Hölder continuous of any order up to $1/(1+\kappa/8)$.

From this, we obtain that the Young integral is well defined along such $\mathrm{SLE}_\kappa$ paths with probability one, and hence that $\mathrm{SLE}_\kappa$ admits a path-wise notion of integration.  This allows us to consider the expected signature of $\mathrm{SLE}$, as defined in rough path theory, and to give a precise formula for its first three gradings.

The main technical result required is a uniform bound on the probability that an $\mathrm{SLE}_\kappa$ crosses an annulus $k$-distinct times.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-21

Publication Date: September 25, 2012

DOI: 10.1214/EJP.v17-2331


  • Aizenman, M.; Burchard, A. Hölder regularity and dimension bounds for random curves. Duke Math. J. 99 (1999), no. 3, 419--453. MR1712629
  • Beffara, Vincent. The dimension of the SLE curves. Ann. Probab. 36 (2008), no. 4, 1421--1452. MR2435854
  • Bertrand Duplantier and Scott Sheffield, Schramm-Loewner Evolution and Liouville Quantum Gravity, Phys. Rev. Lett. 107 (2011), 131305.
  • Thomas Fawcett, Problems in stochastic analysis: Connections between rough paths and non-commutative harmonic analysis., Ph.D. thesis, Oxford Universy, UK, 2003.
  • Laurence Field, 2011, personal communication.
  • Hambly, Ben; Lyons, Terry. Uniqueness for the signature of a path of bounded variation and the reduced path group. Ann. of Math. (2) 171 (2010), no. 1, 109--167. MR2630037
  • Johansson Viklund, Fredrik; Lawler, Gregory F. Optimal Hölder exponent for the SLE path. Duke Math. J. 159 (2011), no. 3, 351--383. MR2831873
  • Lawler, Gregory F. Conformally invariant processes in the plane. Mathematical Surveys and Monographs, 114. American Mathematical Society, Providence, RI, 2005. xii+242 pp. ISBN: 0-8218-3677-3 MR2129588
  • Lawler, Gregory F. Conformal invariance and 2D statistical physics. Bull. Amer. Math. Soc. (N.S.) 46 (2009), no. 1, 35--54. MR2457071
  • Lawler, Gregory F. Schramm-Loewner evolution (SLE), Statistical mechanics, IAS/Park City Math. Ser., vol. 16, Amer. Math. Soc., Providence, RI, 2009, pp. 231--295. MR2523461 (2011d:60244)
  • Lawler, Gregory F.; Schramm, Oded; Werner, Wendelin. Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004), no. 1B, 939--995. MR2044671
  • Lawler, Gregory F.; Sheffield, Scott. A natural parametrization for the Schramm-Loewner evolution. Ann. Probab. 39 (2011), no. 5, 1896--1937. MR2884877
  • Gregory F. Lawler and Brent M. Werness, Multi-point Green's functions for SLE and an estimate of Beffara, 2011, arXiv:1011.3551v2. To appear in The Annals of Probability.
  • Gregory F. Lawler and Wang Zhou, SLE curves and natural parametrization, 2010, arXiv:1006.4936. To appear in The Annals of Probability.
  • Lind, Joan R. Hölder regularity of the SLE trace. Trans. Amer. Math. Soc. 360 (2008), no. 7, 3557--3578. MR2386236
  • Pierre-Louis Lions, Remarques sur l'intégration et les équations différentielles ordinaires, 2009, Lecture available at:
  • Terry Lyons and Hao Ni, Expected signature of two dimensional Brownian Motion up to the first exit time of the domain, 2011, arxiv:1101.5902v2.
  • Lyons, Terry J.; Caruana, Michael; Lévy, Thierry. Differential equations driven by rough paths. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004. With an introduction concerning the Summer School by Jean Picard. Lecture Notes in Mathematics, 1908. Springer, Berlin, 2007. xviii+109 pp. ISBN: 978-3-540-71284-8; 3-540-71284-4 MR2314753
  • Peter Mörters and Yuval Peres, Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, Cambridge, 2010, With an appendix by Oded Schramm and Wendelin Werner. MR2604525 (2011i:60152)
  • Rohde, Steffen; Schramm, Oded. Basic properties of SLE. Ann. of Math. (2) 161 (2005), no. 2, 883--924. MR2153402
  • Schramm, Oded. Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221--288. MR1776084
  • Schramm, Oded. A percolation formula. Electron. Comm. Probab. 6 (2001), 115--120 (electronic). MR1871700
  • Schramm, Oded; Sheffield, Scott. Contour lines of the two-dimensional discrete Gaussian free field. Acta Math. 202 (2009), no. 1, 21--137. MR2486487
  • Smirnov, Stanislav. Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 3, 239--244. MR1851632
  • Smirnov, Stanislav. Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. of Math. (2) 172 (2010), no. 2, 1435--1467. MR2680496
  • Werner, Wendelin. Random planar curves and Schramm-Loewner evolutions. Lectures on probability theory and statistics, 107--195, Lecture Notes in Math., 1840, Springer, Berlin, 2004. MR2079672
  • Young, L. C. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936), no. 1, 251--282. MR1555421

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.