Gaussian generalized random processes on \({\mathcal K}\{M_ p\}\) spaces (Q1323073)

The authors give representation theorems for generalized Gaussian random processes on a class of \(K\{M_ p\}\) spaces of Gelfand and Shilov. One such representation theorem as proved by them is: \[ \xi (w, \varphi) = \sum_{| \alpha | \leq p} \int_{R^ n} f_ \alpha (w,t) M_ p(t) D^ \alpha \varphi (t)dt, \quad \forall \varphi \in K, \] where \(\xi\) is a Gaussian g.r.p. on \(\Omega \times K\) and \(f_ \alpha (w,t)\) are Gaussian random processes on \(\Omega \times R^ n\) such that for every \(w \in \Omega\), \(f_ \alpha (w, \cdot) \in L^ 2 (R^ n)\). Also, \[ \bigl | \xi (w, \varphi) \bigr | \leq c(w) \| \varphi \|_{p.2}, \quad w \in \Omega, \;\varphi \in K, \] and some additional conditions are satisfied by \(\xi (w, \varphi)\).