Generalized Wright Analysis in Infinite Dimensions

Abstract

This paper investigates a broad class of non-Gaussian measures, $ \mu_\Psi$, associated with a family of generalized Wright functions, $_m\Psi_q$. First, we study these measures in Euclidean spaces $\mathbb{R}^d$, then define them in an abstract nuclear triple $\mathcal{N}\subset\mathcal{H}\subset\mathcal{N}'$. 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 $L^2(\mu_\Psi)$ to the nuclear triple consisting of test functions' and distributions' spaces, $(\mathcal{N})^{1}\subset L^2(\mu_\Psi)\subset(\mathcal{N})_{\mu_\Psi}^{-1}$. Furthermore, thanks to the definition of two transformations, $S_{\mu_{\Psi}}$ and $T_{\mu_{\Psi}}$, we study Donsker's delta as an element within $(\mathcal{N})_{\mu_\Psi}^{-1}$ applying the integral equations fulfilled by $_m\Psi_q$.