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

TL;DR

This paper demonstrates that SLE paths can be reparametrized to achieve optimal Hölder continuity, enabling path-wise integration and analysis of their signatures within rough path theory.

Contribution

It establishes the Hölder regularity of SLE paths for kappa ≤ 4, allowing for path-wise integration and explicit signature formulas, advancing the understanding of SLE in rough path context.

Findings

SLE paths are reparametrizable to be Hölder continuous up to a specific order.

Young integrals are well-defined along SLE paths with probability one.

Provides a formula for the expected signature of SLE in rough path theory.

Abstract

When studying stochastic processes, it is often fruitful to have an understanding of several different notions of regularity. One such notion is the optimal H\"older exponent obtainable under reparametrization. In this paper, we show that the chordal SLE_kappa path in the unit disk for kappa less than or equal to 4 can be reparametrized to be H\"older continuous of any order up to 1/(1+kappa/8). From this result, we obtain that the Young integral is well defined along such SLE paths with probability one, and hence that SLE admits a path-wise notion of integration. This allows for us to consider the expected signature of 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 a SLE crosses an annulus k-distinct times.