Splitting algorithm and normed convergence for drawing the random Loewner curves

TL;DR

This paper introduces a new splitting algorithm for simulating random Loewner curves, with convergence guarantees, and extends it to fractional and noise-reinforced SLE, offering insights into fractal dimensions and statistical properties.

Contribution

A novel splitting algorithm with rigorous convergence analysis for simulating random Loewner curves, extended to fractional and noise-reinforced SLE models.

Findings

Algorithm achieves convergence in sup-norm and L^p.

Extensions enable predictions of fractal dimensions.

Provides new insights into statistical properties of SLE variants.

Abstract

Recent advances in Schramm-Loewner evolution have driven increasing interest in non-standard Loewner flows. In this work, we propose a novel splitting algorithm to simulate random Loewner curves with rigorous convergence analysis in sup-norm and $L^p$. The algorithm is further extended to explore fractional SLE, driven by fractional Brownian motion, and noise-reinforced SLE, incorporating the effect on long-term memory. These exploratory and numerical extensions enable theoretical predictions on fractal dimensions and other statistical phenomena, providing new insights into such dynamics and opening directions for future research.