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


 
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1. Title Title of document Regularity of Schramm-Loewner evolutions, annular crossings, and rough path theory
 
2. Creator Author's name, affiliation, country Brent Morehouse Werness; University of Washington; United States
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) Schramm-Loewner Evolutions; H\"older regularity; rough path theory; Young integral; signature
 
3. Subject Subject classification 60J67; 60H05
 
4. Description 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.
 
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7. Date (YYYY-MM-DD) 2012-09-25
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ejp.ejpecp.org/article/view/2331
 
10. Identifier Digital Object Identifier 10.1214/EJP.v17-2331
 
11. Source Journal/conference title; vol., no. (year) Electronic Journal of Probability; Vol 17
 
12. Language English=en en
 
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