Analysis for a class of stochastic fractional nonlinear Schrödinger equations

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.