The main result of this paper is to extend to Hilbert module level the proof of the inclusion of (non-Hamiltonian) stochastic differential equations based on free noise into the class of Hamiltonian equations driven by free white noise. To achieve this goal, free white noise calculus is extended to a trivial Hilbert module. The white noise formulation of the Ito table is radically different from the usual Itô tables, both classical and quantum and, combined with the Accardi–Boukas approach to Ito algebra, allows to drastically simplify calculations. Infinitesimal generators of Hilbert module free flows are characterized in terms of stochastic derivations from an initial algebra into a white noise Itô algebra. We prove that any such derivation is the difference of a ⋆-homomorphism and a trivial embedding.

中文翻译：

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无模块白噪声流

本文的主要成果是将基于自由噪声的（非哈密顿）随机微分方程包含在由自由白噪声驱动的哈密顿方程类中的证明扩展到希尔伯特模块级别。为了实现这个目标，免费的白噪声演算被扩展到一个简单的希尔伯特模块。Ito 表的白噪声公式与通常的 Itô 表（包括经典表和量子表）完全不同，并且与 Ito 代数的 Accardi-Boukas 方法相结合，可以大大简化计算。希尔伯特模自由流的无穷小发生器的特征在于从初始代数到白噪声伊藤代数的随机推导。我们证明任何这样的推导都是 ⋆- 同态和平凡嵌入的区别。