Loewner traces driven by Levy processes

TL;DR

This work extends Loewner evolutions to general Lévy-driven drivers, showing that, under mild regularity (including stable processes), the resulting hulls are almost surely generated by a càdlàg trace and possess strong topological regularity (local connectivity and Hölder-domain complements for κ ≠ 4). The authors develop a robust framework using forward and backward Loewner flows, grids of forward flow estimates, and supermartingale arguments to control derivatives in the presence of jumps, circumventing continuity-based methods. The results recover and extend known cases (e.g., SLEκ with Brownian drivers) and provide a flexible platform for studying dendritic growth and multifractal spectra beyond scale invariance. The findings offer precise conditions under which traces exist and hull complements remain Hölder domains, contributing to our understanding of the geometric complexity of Lévy-driven planar growth models and their domain Markov structure. The work thereby broadens the toolbox for planar random growth models and their potential connections to conformal geometry and statistical physics.

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 $κ\in [0,8)$, then the jump measure of $W$ is required to be locally (upper) Ahlfors regular near the origin, while if $κ> 8$, 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 $κ\neq 4$ (which is not expected to hold when $κ=4$), 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.

Figures (11)

• Figure 1.1: Illustration of the Loewner hull driven by a càdlàg function, made by Toby Cathcart Burn from his GitHub https://github.com/penteract/sle.
• Figure 2.1: Illustration of the maps and hulls in Lemma \ref{['lem: some markov property']}.
• Figure 2.2: Illustration of a path connected but not locally connected comb.
• Figure 3.1: Illustrating quantities in Lemma \ref{['lem: alphaprime and thetaprime']}: Plots of $\kappa \mapsto {\theta_{\kappa,\lambda_\varepsilon}^{\textnormal{h\"ol}}}$ with discrete values $\lambda_\varepsilon = c \, {\lambda_{\kappa}^{\textnormal{h\"ol}}}$ for $c \in \{ 0, \frac{1}{5}, \frac{2}{5} ,\frac{3}{5} , \frac{4}{5} \}$. The largest plot (blue) has $\lambda_\varepsilon = 0$.
• Figure 3.2: Illustrating quantities in Lemma \ref{['lem: alphaprime and thetaprime']}: Plots of $\kappa \mapsto {\lambda_{\kappa}^{\textnormal{h\"ol}}}$.

Theorems & Definitions (134)

• Definition 1.1
• Proposition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: Koebe distortion in $\mathbb{H}$
• proof
• Lemma 2.2
• proof
• Remark 2.3