Electronic Journal of Probability

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

Brent Werness

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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.

Article information

Electron. J. Probab., Volume 17 (2012), paper no. 81, 21 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J67: Stochastic (Schramm-)Loewner evolution (SLE)
Secondary: 60H05: Stochastic integrals

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

This work is licensed under aCreative Commons Attribution 3.0 License.


Werness, Brent. Regularity of Schramm-Loewner evolutions, annular crossings, and rough path theory. Electron. J. Probab. 17 (2012), paper no. 81, 21 pp. doi:10.1214/EJP.v17-2331. https://projecteuclid.org/euclid.ejp/1465062403

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