Loewner traces driven by Levy processes

TL;DR

This paper proves that Loewner chains driven by Levy processes, including stable processes, are almost surely generated by cadlag curves with intricate topological properties, expanding understanding of their geometric and fractal characteristics.

Contribution

It establishes that Levy-driven Loewner chains are generated by cadlag curves and analyzes their topological and geometric properties without assuming regularity, broadening the scope of known results.

Findings

Loewner hulls are a.s. locally connected and path-connected.

Complement of hulls are Holder domains when ppa 0 7 4.

Results rely on derivative estimates and supermartingale domination arguments.

Abstract

Loewner chains with Levy drivers have been proposed as models for random dendritic growth in two dimensions, and as candidates for finding extremal multifractal spectra in problems in classical function theory. These processes are not scale-invariant in general, but they do enjoy a natural domain Markov property thanks to the stationary independent increments of Levy processes. The associated Loewner hulls feature remarkably intricate topological properties, of which very little is known rigorously. We prove that a chordal Loewner chain driven by a Levy process $W$ satisfying mild regularity conditions (including stable processes) is a.s. generated by a cadlag curve. Specifically, if the diffusivity parameter of the driving process $W$ is $\kappa \in [0,8)$, then the jump measure of $W$ is required to be locally (upper) Ahlfors regular near the origin, while if $\kappa > 8$, no constraints are imposed. In particular, we show that the associated Loewner hulls are a.s. locally connected and path-connected. We also show that, the complements of the hulls are a.s. Holder domains when $\kappa \neq 4$ (which is not expected to hold when $\kappa=4$), without any regularity assumptions. The proofs of these results mainly rely on careful derivative estimates for both the forward and backward Loewner maps obtained using delicate but robust enough supermartingale domination arguments. As one cannot control the jump accumulation of general Levy processes, we must circumvent all reasoning that would use continuity. To prove the local connectedness, we use an extension of part of the Hahn-Mazurkiewicz theorem: hulls generated by cadlag curves are locally connected even when jumps would occur at infinite intensity.