On Properties of Statistically Stationary Solutions to the One-Dimensional Schr\"odinger Map Equation

TL;DR

This paper explores the qualitative properties of statistically stationary solutions to the Schr"odinger map equation, revealing their measure-theoretic characteristics and decay properties, with implications for related curvature flow models.

Contribution

It establishes absolute continuity and Gaussian decay of observables' laws, and shows the measure's Hausdorff dimension is at least two for stationary solutions.

Findings

Laws of observables are absolutely continuous with respect to Lebesgue measure.

Energy exhibits Gaussian decay properties.

Stationary solution measures have Hausdorff dimension at least two.

Abstract

We investigate further qualitative properties of statistically stationary solutions to the Schr\"odinger map equation (SME) and the Binormal Curvature Flow (BCF), continuing the work initiated by E. G., M. Hofmanov\'a. Concerning the statistically stationary solutions to the SME, we show that the laws of some relevant observables (such as the space average and the energy) are absolutely continuous with respect to the Lebesgue measure, with a Gaussian decay property for the energy. We further prove that the law $\mu$ of the statistically stationary solution has dimension of at least two: this means that any compact set of Hausdorff dimension smaller than two has $\mu$-measure zero. These properties, with appropriate modifications of the norms, pass directly to the statistically stationary solutions to the BCF.