Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Hölder regularity of the SLE trace

Author: Joan R. Lind
Journal: Trans. Amer. Math. Soc. 360 (2008), 3557-3578
MSC (2000): Primary 60D05, 30C35, 60G17
Published electronically: January 9, 2008
MathSciNet review: 2386236
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: S. Rohde and O. Schramm have recently shown that the SLE trace is Hölder continuous (2005). However, their results are not optimal for all values of $ \kappa$ and only yield a Hölder exponent near $ \frac{1}{2}$ for $ \kappa$ near 0. In this paper, we give improved lower bounds on the optimal Hölder exponent for two natural parametrizations of the SLE trace. Our estimates give a Hölder exponent near 1 for $ \kappa$ near 0, as expected. The work of I. Binder and B. Duplantier (2002) suggests that our results are optimal for the two parametrizations considered.

References [Enhancements On Off] (What's this?)

  • 1. V. Beffara, The dimension of the SLE curves, arXiv:math.PR/0211322.
  • 2. D. Beliaev, Harmonic measure on random fractals, Royal Institute of Technology, Stockholm, 2005.
  • 3. I. Binder and B. Duplantier, Harmonic measure and winding of conformally invariant curves, Physical Review Letters 89 (2002), no. 26.
  • 4. I. Gruzberg and L. Kadanoff, The Loewner equation: maps and shapes, J. Statist. Phys. 114 (2004), no. 5-6, 1183-1198. MR 2039475 (2005f:81210)
  • 5. G. Lawler, O. Schramm, and W. Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237-273. MR 1879850 (2002m:60159a)
  • 6. -, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939-995. MR 2044671 (2005f:82043)
  • 7. J. Lind, A sharp condition for the Loewner equation to generate slits, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 1, 143-158. MR 2140303 (2006b:30013)
  • 8. B. Oksendal, Stochastic differential equations, Springer, 1995. MR 1411679 (97f:60114)
  • 9. S. Rohde and O. Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), no. 2, 883-924. MR 2153402 (2006f:60093)
  • 10. O. Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221-288. MR 1776084 (2001m:60227)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60D05, 30C35, 60G17

Retrieve articles in all journals with MSC (2000): 60D05, 30C35, 60G17

Additional Information

Joan R. Lind
Affiliation: Department of Mathematics, Belmont University, 1900 Belmont Boulevard, Nashville, Tennessee 37212

Received by editor(s): February 24, 2005
Received by editor(s) in revised form: April 22, 2006
Published electronically: January 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society