## Electronic Journal of Probability

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

Brent Werness

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ejp/1465062403

Digital Object Identifier
doi:10.1214/EJP.v17-2331

Mathematical Reviews number (MathSciNet)
MR2981906

Zentralblatt MATH identifier
1252.60084

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

Rights

#### Citation

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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