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2012 Regularity of Schramm-Loewner evolutions, annular crossings, and rough path theory
Brent Werness
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Electron. J. Probab. 17: 1-21 (2012). DOI: 10.1214/EJP.v17-2331


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.


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Brent Werness. "Regularity of Schramm-Loewner evolutions, annular crossings, and rough path theory." Electron. J. Probab. 17 1 - 21, 2012.


Accepted: 25 September 2012; Published: 2012
First available in Project Euclid: 4 June 2016

zbMATH: 1252.60084
MathSciNet: MR2981906
Digital Object Identifier: 10.1214/EJP.v17-2331

Primary: 60J67
Secondary: 60H05

Keywords: H\"older regularity , rough path theory , Schramm-Loewner Evolutions , signature , Young integral


Vol.17 • 2012
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