Hölder regularity of the SLE trace

Lind, Joan

doi:10.1090/S0002-9947-08-04327-4

Hölder regularity of the SLE trace
HTML articles powered by AMS MathViewer

by Joan R. Lind PDF

Trans. Amer. Math. Soc. 360 (2008), 3557-3578 Request permission

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

V. Beffara, The dimension of the SLE curves, arXiv:math.PR/0211322.
D. Beliaev, Harmonic measure on random fractals, Royal Institute of Technology, Stockholm, 2005.
I. Binder and B. Duplantier, Harmonic measure and winding of conformally invariant curves, Physical Review Letters 89 (2002), no. 26.
Ilya A. Gruzberg and Leo P. Kadanoff, The Loewner equation: maps and shapes, J. Statist. Phys. 114 (2004), no. 5-6, 1183–1198. MR 2039475, DOI 10.1023/B:JOSS.0000013973.40984.3b
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), no. 2, 237–273. MR 1879850, DOI 10.1007/BF02392618
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Ann. Probab. 32 (2004), no. 1B, 939–995. MR 2044671, DOI 10.1214/aop/1079021469
Joan R. Lind, A sharp condition for the Loewner equation to generate slits, Ann. Acad. Sci. Fenn. Math. 30 (2005), no. 1, 143–158. MR 2140303
Bernt Øksendal, Stochastic differential equations, 4th ed., Universitext, Springer-Verlag, Berlin, 1995. An introduction with applications. MR 1411679, DOI 10.1007/978-3-662-03185-8
Steffen Rohde and Oded Schramm, Basic properties of SLE, Ann. of Math. (2) 161 (2005), no. 2, 883–924. MR 2153402, DOI 10.4007/annals.2005.161.883
Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel J. Math. 118 (2000), 221–288. MR 1776084, DOI 10.1007/BF02803524