Generalized Wright Analysis in Infinite Dimensions

This paper investigates a broad class of non-Gaussian measures, ${\mu }_{\mathrm{\Psi }}$, associated with a family of generalized Wright functions, ${}_m{\mathrm{\Psi }}_q$. First, we study these measures in Euclidean spaces ${\mathbb{R}}^d$, then define them in an abstract nuclear triple $\mathcal{N}\subset \mathcal{H}\subset {\mathcal{N}}^{\mathrm{\prime }}$. We study analyticity, invariance properties, and ergodicity under a particular group of automorphisms. Then we show the existence of an Appell system which allows the extension of the non-Gaussian Hilbert space $L^2({\mu }_{\mathrm{\Psi }})$ to the nuclear triple consisting of test functions’ and distributions’ spaces, $(\mathcal{N}{)}^1\subset L^2({\mu }_{\mathrm{\Psi }})\subset (\mathcal{N}{)}_{{\mu }_{\mathrm{\Psi }}}^{-1}$. Furthermore, thanks to the definition of two transformations, $S_{{\mu }_{\mathrm{\Psi }}}$ and $T_{{\mu }_{\mathrm{\Psi }}}$, we study Donsker’s delta as an element within $(\mathcal{N}{)}_{{\mu }_{\mathrm{\Psi }}}^{-1}$ applying the integral equations fulfilled by ${}_m{\mathrm{\Psi }}_q$.