We adapt Lyon’s rough path theory to study Landau–Lifshitz–Gilbert equations (LLGEs) driven by geometric rough paths in one dimension, with non-zero exchange energy only. We convert the LLGEs to a fully nonlinear time-dependent partial differential equation without rough paths term by a suitable transformation. Our point of interest is the regular approximation of the geometric rough path. We investigate the limit equation, the form of the correction term, and its convergence rate in controlled rough path spaces. The key ingredients for constructing the solution and its corresponding convergence results are the Doss–Sussmann transformation, maximal regularity property, and the geometric rough path theory.

我们采用 Lyon 的粗糙路径理论来研究由一维几何粗糙路径驱动的 Landau-Lifshitz-Gilbert 方程 (LLGE)，仅具有非零交换能量。我们通过适当的变换将 LLGE 转换为完全非线性的时间相关偏微分方程，而没有粗糙路径项。我们的兴趣点是几何粗糙路径的规则近似。我们研究了极限方程、修正项的形式及其在受控粗糙路径空间中的收敛速度。构建解及其相应收敛结果的关键要素是 Doss-Sussmann 变换、最大正则性和几何粗糙路径理论。