We study the stochastic Nonlinear Schrodinger system with multiplicative white noise in energy space $H^1$. Based on deterministic and stochastic Strichartz estimates, we prove the local well-posedness and uniqueness of mild solution. Then we prove the global well-posedness in the mass subcritical case and the defocusing case. For the mass subcritical case, we also investigate the global existence when the $L^2$ norm of initial value is small enough. In addition, we study the blow-up phenomenon and give a sharp criteria.

中文翻译：

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随机非线性薛定谔系统的全局解和爆破

我们研究了能量空间 $H^1$ 中具有乘法白噪声的随机非线性薛定谔系统。基于确定性和随机 Strichartz 估计，我们证明了温和解的局部适定性和唯一性。然后我们证明了大众次临界情况和散焦情况下的全局适定性。对于质量次临界情况，我们还研究了当初始值的 $L^2$ 范数足够小时的全局存在性。此外，我们研究了爆破现象并给出了一个明确的标准。