arXiv subjects
Publications and source records attributed to Steffen Rohde.
We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of the hulls are Hoelder domains, prove that the hulls have Hausdorff dimension 1, and show that the trace is right-continuous with left limits almost surely.
In the early 1980's an elementary algorithm for computing conformal maps was discovered by R. Kühnau and the first author. The algorithm is fast and accurate, but convergence was not known. Given points z_0,...,z_n in the plane, the algorithm computes an explicit conformal map of the unit disk onto a region bounded by a Jordan curve γwith z_0,...,z_n \in γ. We prove convergence for Jordan regions in the sense of uniformly close boundaries, and give corresponding uniform estimates on the closed region and the closed disc for the mapping functions and their inverses. Improved estimates are obtained if the data points lie on a C^1 curve or a K-quasicircle. The algorithm was discovered as an approximate method for conformal welding, however it can also be viewed as a discretization of the Löwner differential equation.
SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $κ$. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions. The present paper attempts a first systematic study of SLE. It is proved that for all $κ\ne 8$ the SLE trace is a path; for $κ\in[0,4]$ it is a simple path; for $κ\in(4,8)$ it is a self-intersecting path; and for $κ>8$ it is space-filling. It is also shown that the Hausdorff dimension of the SLE trace is a.s. at most $1+κ/8$ and that the expected number of disks of size $\eps$ needed to cover it inside a bounded set is at least $\eps^{-(1+κ/8)+o(1)}$ for $κ\in[0,8)$ along some sequence $\eps\to 0$. Similarly, for $κ\ge 4$, the Hausdorff dimension of the outer boundary of the SLE hull is a.s. at most $1+2/κ$, and the expected number of disks of radius $\eps$ needed to cover it is at least $\eps^{-(1+2/κ)+o(1)}$ for a sequence $\eps\to 0$.
A uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains is established. Exact Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are given. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also mentioned.
We prove that the Julia set of a rational map of the Riemann sphere satisfying the Collet-Eckmann condition and having no parabolic periodic point is mean porous, if it is not the whole sphere. It follows that the Minkowski dimension of the Julia set is less than 2.