We construct a martingale solution of the stochastic nonlinear Schrödinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in \(H^1\) on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood–Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of càdlàg functions and Jakubowski’s generalization of the Skorohod-Theorem to nonmetric spaces.

中文翻译：

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由纯跳跃噪声驱动的随机非线性Schrödinger方程的弱mar解决方案

我们以Marcus典范形式构造具有跳跃型乘性噪声的随机非线性Schrödinger方程（NLS）的mar解决方案。该问题是在一个通用框架中提出的，该框架涵盖了紧凑流形和具有各种边界条件的有界域上\（H ^ 1 \）中的亚临界聚焦和散焦随机NLS 。该证明基于Faedo-Galerkin方法的一种变体。在近似方程的表述中，源自Littlewood-Paley分解的有限维算符对经典的正交投影进行了补充，以保证统一的估计。构造的其他组成部分是某些càdlàg函数空间的紧度标准以及Jakubowski将Skorohod-Theorem推广到非度量空间。