本项目旨在研究在一维和三维区域上的随机Landau-Lifshitz-Gilbert（SLLG）方程。.LLG方程本身描述了铁磁体上的磁化强度随时空变化的演化规律。铁磁性的应用对现代工业的发展起了极为重要的作用。对于确定性的LLG方程，国内外已有大量的研究成果。.对于一维情形，本项目拟证明满足周期性边值条件的一维有界区域上SLLG方程强解的存在唯一性并研究其小扰动渐近行为，特别是证明其准势就是方程的能量，并且证明当SLLG方程噪声的相关半径趋于0时，其准势会收敛到相对应的确定性方程的准势。目前在国内外只有很少对于SLLG小扰动渐近行为的研究，特别是还没有对其准势的研究。.对于三维情形，本项目拟证明满足非齐次Neumann边值条件的三维连通与非连通有界区域上的SLLG方程与全空间上的Maxwell方程联立后其弱解的存在性。对于此问题，目前国内外只有对确定性方程的结果。

We would like to study the one dimensional and three dimensional stochastic Landau-Lifshitz-Gilbert (SLLG) equations..The LLG equation describes the evolution of the magnetization in ferromagnetic materials. The applications of ferromagnetic materials are very important in modern industrial world. There are already a lot of research on the deterministic LLG equations. .For the one dimensional case, we are going to study the existence and uniqueness of the strong solution of the SLLG equations in a bounded domain with periodic boundary conditions and suitable initial conditions. Moreover, we would like to study the small noise asymptotic behavior of the solution. In particular, we will show that the quasipotential is the energy of the equation and prove that as the correlation radius of the noise tends to 0, the quasipotential of the SLLG will converge to the quasipotential of the deterministic LLG. So far as we know there are only a few work on the small noise asymptotic behavior for the SLLG and no results on the study of quasipotential. .For the three dimensional case, we are going to study the existence of weak solutions of the SLLG equations in a connected or unconnected bounded domain with inhomogenous Neumann boundary conditions, coupled with the Maxwell equations in the whole space. So far as we know, there are only such results for the deterministic LLG equations.