White noise differential equations for vector-valued white noise functionals

Abstract

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.