In this article, we construct a global martingale solution to a general nonlinear Schrödinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like \({{\mathbb {R}}^d}\), non-compact Riemannian manifolds, and unbounded domains in \({{\mathbb {R}}^d}\) with different boundary conditions. The initial value belongs to the energy space \(H^1\) and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood–Paley-decomposition. In the limit procedure, we employ tightness of the approximated solutions and Jakubowski’s extension of the Skorohod theorem to nonmetric spaces.

在本文中，我们为Stratonovich形式的线性线性乘性噪声构造了一个通用非线性Schrödinger方程的全局mar解。我们的框架包括许多空间域的示例，例如\（{{\ mathbb {R}} ^ d} \），非紧致黎曼流形和\（{{\ mathbb {R}} ^ d} \）中的无界域具有不同的边界条件。初始值属于能量空间\（H ^ 1 \）我们将处理亚临界聚焦和散焦功率非线性。该证明基于一种近似技术，该技术利用了频谱理论方法和抽象的Littlewood-Paley分解。在极限过程中，我们采用了近似解的紧密性，并将雅各博夫斯基的Skorohod定理扩展到非度量空间。