Generalized Wright Analysis in Infinite Dimensions

Abstract

This paper investigates a broad class of non-Gaussian measures, $\hat{I}{¼}_{\hat{I}}¨$, associated with a family of generalized Wright functions, ${}_m\hat{I}{¨}_q$. First, we study these measures in Euclidean spaces ${\mathbb{R}}^d$, then define them in an abstract nuclear triple $\mathcal{N}\subset \mathcal{H}\subset {\mathcal{N}}^{\prime }$. 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 $L^2(\hat{I}{¼}_{\hat{I}}¨)$ to the nuclear triple consisting of test functions' and distributions' spaces, $(\mathcal{N}{)}^1\subset L^2(\hat{I}{¼}_{\hat{I}}¨)\subset (\mathcal{N}{)}_{\hat{I}{¼}_{\hat{I}}¨}^{-1}$. Furthermore, thanks to the definition of two transformations, $S_{\hat{I}{¼}_{\hat{I}}¨}$ and $T_{\hat{I}{¼}_{\hat{I}}¨}$, we study Donsker's delta as an element within $(\mathcal{N}{)}_{\hat{I}{¼}_{\hat{I}}¨}^{-1}$ applying the integral equations fulfilled by ${}_m\hat{I}{¨}_q$.