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.
Highlights
Oded Schramm introduced Schramm-Loewener Evolutions (SLE) as a stochastic process to serve as the scaling limit of various discrete models from statistical physics believed to be conformally invariant in the limit [21]
It has successfully been used to study a number of such processes
It is conjectured that SLEκ under the natural parametrization should have the optimal Hölder exponent our result indicates our techniques do not immediately illuminate this question
Summary
Oded Schramm introduced Schramm-Loewener Evolutions (SLE) as a stochastic process to serve as the scaling limit of various discrete models from statistical physics believed to be conformally invariant in the limit [21]. The critical case of α = 1/d is still open, it is natural to conjecture that it cannot be reparametrized to be Hölder continuous of this order With this result, we are able to provide a few preliminary results in the rough path theory of SLE. We obtain a definition of integration against a SLEκ curve This result shows that SLEκ for κ ≤ 4 has finite d-variation for some d < 2 in the sense used in [18] and both the Young integral and the integral of Lions as defined in [16] give a way of almost surely integrating path-wise along an SLE curve.
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