On Properties of Statistically Stationary Solutions to the One-Dimensional Schrödinger Map Equation.

Abstract We investigate further qualitative properties of statistically stationary solutions to the Schrödinger map equation (SME) and the Binormal Curvature Flow (BCF) on a bounded interval, continuing the work initiated by E. G., M. Hofmanová [25]. 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 $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> of the statistically stationary solution has dimension of at least two: this means that any compact set of Hausdorff dimension smaller than two has $$\mu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> </mml:math> -measure zero. These properties, with appropriate modifications of the norms, pass directly to the statistically stationary solutions to the BCF.