The stochastic fractional nonlinear Schr\"odinger equations in $H^{\alpha}$ and structure-preserving algorithm

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.