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

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 SLEκ in the unit disk for κ ≤ 4 can be reparametrized to be Hölder continuous of any order up to 1/(1 + κ/8). From this, we obtain that the Young integral is well defined along such SLEκ paths with probability one, and hence that SLEκ admits a path-wise notion of integration. This allows 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 an SLEκ crosses an annulus k-distinct times.