We analyze the qualitative properties and the order of convergence of a splitting scheme for a class of nonlinear stochastic Schr\"odinger equations driven by additive It\^o noise. The class of nonlinearities of interest includes nonlocal interaction cubic nonlinearities. We show that the numerical solution is symplectic and preserves the expected mass for all times. On top of that, for the convergence analysis, some exponential moment bounds for the exact and numerical solutions are proved. This enables us to provide strong orders of convergence as well as orders of convergence in probability and almost surely. Finally, extensive numerical experiments illustrate the performance of the proposed numerical scheme.

中文翻译：

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一类非线性随机Schr\"odinger方程的分裂格式分析

我们分析了一类由加性 It\^o 噪声驱动的非线性随机 Schr\"odinger 方程的分裂方案的定性性质和收敛阶数。感兴趣的非线性类包括非局部相互作用三次非线性。我们表明数值解是辛的并且始终保持预期质量。除此之外，为了收敛分析，证明了精确解和数值解的一些指数矩界。这使我们能够提供强收敛阶以及概率收敛，几乎可以肯定。最后，大量的数值实验说明了所提出的数值方案的性能。