Abstract:

Based on a Gel’fand triple $(\mathcal{N})\otimes \mathcal{E}\subset \mathrm{\Gamma }(H)\otimes \mathfrak{h}\subset ((\mathcal{N})\otimes \mathcal{E}{)}^{\ast }$, we introduce a new notion of Wick type product of generalized Gaussian white noise functionals which is associated with a continuous bilinear mapping $\mathfrak{B}:{\mathcal{E}}^{\ast }\times {\mathcal{E}}^{\ast }\to {\mathcal{E}}^{\ast }$. Then we study Wick type differential equations for vector-valued generalized Gaussian white noise functionals and, as a simple application, we study Wick type differential equations for matrix-valued generalized Gaussian white noise functionals. For our purposes, we make a systematic study of equicontinuity of the left and right Wick type multiplication operators.

2010 AMS Mathematics Subject Classification: Primary 60H40; Secondary 46F25, 46A11.

Keywords and phrases: white noise theory, Gaussian white noise functional, Wick product, equicontinuous, Wick type differential equation.