We 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 $H^{\alpha}$. Using a variational principle, we demonstrate 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. We develop a structure-preserving algorithm for the stochastic fractional nonlinear Schr\"odinger equation from the perspective of symplectic geometry. It is established that the stochastic midpoint scheme satisfies the corresponding symplectic law in the discrete sense. Furthermore, since the midpoint scheme is implicit, we also develop a more effective mass-preserving splitting scheme. Consequently, the convergence order of the splitting scheme is shown to be $1$. Two numerical examples are conducted to validate the efficiency of the theory.

翻译：我们研究一类在合适能量空间$H^{\alpha}$中具有径向对称初值的随机分数阶非线性薛定谔方程解的全局存在性。利用变分原理，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。我们从辛几何角度为随机分数阶非线性薛定谔方程发展了一种保结构算法。证明随机中点格式在离散意义上满足相应的辛律。此外，由于中点格式是隐式的，我们还发展了一种更有效的质量守恒分裂格式。由此，该分裂格式的收敛阶被证明为$1$。通过两个数值算例验证了理论的有效性。