Weak Martingale Solutions of the Stochastic Schrödinger-Poisson-Landau-Lifshitz-Gilbert System
Yurong Wei, Huaqiao Wang
TL;DR
This work addresses the existence of weak martingale solutions for the three-dimensional stochastic SPLLG system, which couples Schrödinger–Poisson spin dynamics with Landau–Lifshitz–Gilbert magnetization under both continuous and small jump noise. The authors develop a three-layer approximation scheme—Faedo–Galerkin in the first layer, a penalization in the second layer, and domain-extension in the third—to overcome nonlinearity and coupling challenges. They derive uniform energy bounds, establish tightness via Aldous-type criteria, and employ the Skorokhod–Jakubowski framework to pass to the limit, proving the existence of weak martingale solutions on bounded domains and extending to the whole space. The results generalize prior stochastic LLG and SPLLG analyses by incorporating strong coupling and Lévy noise, providing a rigorous foundation for stochastic spin transport models with saturation constraints. The methods and energy estimates offer a robust blueprint for analyzing similarly structured stochastic multi-physics systems.
Abstract
The Schrödinger-Poisson-Landau-Lifshitz-Gilbert (SPLLG) system can characterize the spin transfer torque mechanism transferring the spin angular momentum to the magnetization dynamics through spin-magnetization coupling. We study the three-dimensional stochastic SPLLG system driven by a multiplicative stochastic force containing a continuous noise and a small jump noise. We establish the existence of weak martingale solutions based on the penalized functional technique, the Faedo-Galerkin approximation, stochastic compactness method, and a careful identification of the limit. Due to the strong coupling and strong nonlinearity caused by the SPLLG system and stochastic effects, some crucial difficulties have been encountered in obtaining energy estimates and avoiding non-negativity of the test function. We mainly utilize the structure of equations and the property of martingales developing the new energy estimates, and apply the three-layer approximation to overcome these difficulties. In particular, we extend the results by Z. Brzeźniak and U. Manna (Comm. Math. Phys., 2019) and by L.H. Chai, C.J. García-Cervera and X. Yang (Arch. Ration. Mech. Anal., 2018) to both the stochastic case and the coupling case.