Abstract:

Based on a Gel’fand triple \((\mathcal{N})\otimes \mathcal{E} \subset \Gamma(H)\otimes \mathfrak{h} \subset ((\mathcal{N})\otimes \mathcal{E})^*\), we introduce a new notion of Wick type product of generalized Gaussian white noise functionals which is associated with a continuous bilinear mapping \(\mathfrak{B}:\mathcal{E}^*\times \mathcal{E}^*\to\mathcal{E}^*\). 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.