We consider the defocusing nonlinear Schrödinger equations on the two-dimensional compact Riemannian manifold without boundary or a bounded domain in \(\mathbb {R}^2\). Our aim is to give a pedagogic and self-contained presentation on the Wick renormalization in terms of the Hermite polynomials and the Laguerre polynomials and construct the Gibbs measures corresponding to the Wick ordered Hamiltonian. Then, we construct global-in-time solutions with initial data distributed according to the Gibbs measure and show that the law of the random solutions, at any time, is again given by the Gibbs measure.

中文翻译：

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行人方法求解二维散焦非线性Schrödinger方程的不变Gibbs测度

我们考虑在\（\ mathbb {R} ^ 2 \）中没有边界或有界域的二维紧凑黎曼流形上的散焦非线性Schrödinger方程。我们的目的是根据Hermite多项式和Laguerre多项式对Wick重整化进行教学和独立的介绍，并构造与Wick有序哈密顿量相对应的Gibbs测度。然后，我们根据Gibbs测度构造具有初始数据分布的全局时间解，并证明随机解的定律在任何时候都由Gibbs测度给出。