In this paper, we first investigate the global existence of a solution for the stochastic fractional nonlinear Schr\"odinger equation with radially symmetric initial data in a suitable energy space $H^{\alpha}$. We then show that the stochastic fractional nonlinear Schr\"odinger equation in the Stratonovich sense forms an infinite-dimensional stochastic Hamiltonian system, with its phase flow preserving symplecticity. Finally, we develop a stochastic midpoint scheme for the stochastic fractional nonlinear Schr\"odinger equation from the perspective of symplectic geometry. It is proved that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. A numerical example is conducted to validate the efficiency of the theory.

翻译：本文首先研究了具有径向对称初始条件的随机分数阶非线性薛定谔方程在适当能量空间$H^{\alpha}$中解的全局存在性。随后，我们证明Stratonovich意义下的随机分数阶非线性薛定谔方程构成一个无限维随机哈密顿系统，其相流保持辛结构。最后，从辛几何视角出发，我们为该方程发展了随机中点格式。理论证明表明，该随机中点格式在离散意义下满足相应的辛律。数值算例验证了该理论的有效性。