Generalized Wright Analysis in Infinite Dimensions

TL;DR

This paper explores non-Gaussian measures linked to generalized Wright functions, analyzing their properties in Euclidean and abstract spaces, and extending Hilbert space frameworks to include distributions and delta functions.

Contribution

It introduces a new class of non-Gaussian measures based on Wright functions, extending the Hilbert space framework to nuclear triples and analyzing their invariance and ergodic properties.

Findings

Established analyticity and invariance of the measures

Constructed an Appell system for the measures

Extended the Hilbert space to include distributions and delta functions

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 $_m\Psi_q$.