Wong-Zakai approximation for Landau-Lifshitz-Gilbert equation with anisotropy energy driven by geometric rough paths

Abstract

We investigate the one-dimensional Rough Landau–Lifshitz–Gilbert Equation (RLLGE) in the presence of nonzero exchange and anisotropy energies, using Lyons' rough path theory. The solutions are constrained to lie on the two-dimensional unit sphere $S^2\subset R^3$, and we prove the existence and uniqueness of strong solutions within this geometric setting. Since the equation evolves on a manifold, a central difficulty arises in approximating geometric rough paths in a regular and controlled manner. We conduct a detailed analysis of the limiting equation, the associated correction term, and its convergence rate in the controlled rough path framework. The construction of solutions and the convergence analysis rely on several key techniques: the Doss–Sussmann transformation, maximal regularity results, and the theory of geometric rough paths. Together, these tools ensure a rigorous treatment of the problem and allow us to capture the essential rough structure of the dynamics.