In this paper, a framework of vector-valued white noise functionals has been constructed as a Gel’fand triple $${\mathcal {W}}_{\alpha }\otimes {\mathcal {E}} \subset \varGamma (H)\otimes K \subset ( {\mathcal {W}}_{\alpha }\otimes {\mathcal {E}})^*$$ . Base on the Gel’fand triple, a new notion of Wick product of vector-valued white noise functionals induced by a bilinear mapping $${\mathfrak {B}}:{\mathcal {E}}^*\times {\mathcal {E}}^*\rightarrow {\mathcal {E}}$$ is introduced and called a $${\mathfrak {B}}$$ -Wick product. We establish the unique existence of solution of an abstract functional differential equation base on the Gel’fand triple. For our purpose, we improve slightly the well-known analytic characterization theorem for S-transform, and the convergence theorem in terms of S-transform for a sequence of vector-valued generalized white noise functionals. As an application, we study the Wick type differential equations for vector-valued white noise functionals, and as examples we discuss the Wick type differential equations for certain operator algebra valued white noise functionals which naturally includes random matrices of white noise functionals.

中文翻译：

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矢量值白噪声泛函的白噪声微分方程

在本文中，向量值白噪声泛函的框架被构建为 Gel'fand 三元组 $${\mathcal {W}}_{\alpha }\otimes {\mathcal {E}} \subset \varGamma ( H)\otimes K \subset ( {\mathcal {W}}}_{\alpha }\otimes {\mathcal {E}})^*$$ 。基于 Gel'fand 三元组，由双线性映射引起的向量值白噪声泛函的 Wick 乘积的新概念 $${\mathfrak {B}}:{\mathcal {E}}^*\times {\mathcal {E}}^*\rightarrow {\mathcal {E}}$$ 被引入并称为 $${\mathfrak {B}}$$ -Wick 乘积。我们建立了基于 Gel'fand 三元组的抽象泛函微分方程解的唯一存在性。出于我们的目的，我们稍微改进了著名的 S 变换解析表征定理，以及基于向量值广义白噪声泛函序列的 S 变换的收敛定理。作为应用，我们研究向量值白噪声泛函的 Wick 型微分方程，并作为示例讨论某些算子代数值白噪声泛函的 Wick 型微分方程，其中自然包括白噪声泛函的随机矩阵。