Complex Generalized Integral Means Spectrum of Drifted Whole-Plane SLE and LLE

TL;DR

This paper investigates the complex generalized integral means spectrum for whole-plane Loewner evolutions driven by Lévy processes, extending understanding of SLE with drift and symmetric Lévy processes through new theoretical results.

Contribution

It introduces the complex generalized integral means spectrum for Lévy-driven Loewner evolutions, generalizing previous results and connecting to Liouville quantum gravity.

Findings

Derived new formulas for the spectrum with drift

Extended results to symmetric Lévy processes

Connected spectrum analysis to Liouville quantum gravity

Abstract

We present new results for the complex generalized integral means spectrum for two kinds of whole-plane Loewner evolutions driven by L\'evy processes: - L\'evy processes with continuous trajectories, which correspond to Schramm-Loewner evolutions (SLE) with a drift term in the Brownian driving function. A natural path to access the standard integral means spectrum in the presence of drift goes through the introduction of the complex generalized integral means spectrum, which is obtained via the so-called Liouville quantum gravity. -Symmetric L\'evy processes for which we generalize recent results by Loutsenko and Yermolayeva.