Fractional Poisson Analysis in Dimension one

TL;DR

This paper develops a biorthogonal approach to analyze fractional Poisson measures, constructing test and generalized function spaces, and extending Wick calculus from Gaussian to non-Gaussian settings.

Contribution

It introduces a new framework for fractional Poisson measures using Appell systems, characterizing associated function spaces and extending Wick calculus.

Findings

Constructed test and generalized function spaces for fractional Poisson measures.

Established dense embeddings of these spaces within L^2 spaces.

Extended Wick calculus to non-Gaussian fractional Poisson frameworks.

Abstract

In this paper, we use a biorthogonal approach (Appell system) to construct and characterize the spaces of test and generalized functions associated to the fractional Poisson measure $\pi_{\lambda,\beta}$, that is, a probability measure in the set of natural (or real) numbers. The Hilbert space $L^{2}(\pi_{\lambda,\beta})$ of complex-valued functions plays a central role in the construction, namely, the test function spaces $(N)_{\pi_{\lambda,\beta}}^{\kappa}$, $\kappa\in[0,1]$ is densely embedded in $L^{2}(\pi_{\lambda,\beta})$. Moreover, $L^{2}(\pi_{\lambda,\beta})$ is also dense in the dual $((N)_{\pi_{\lambda,\beta}}^{\kappa})'=(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. Hence, we obtain a chain of densely embeddings $(N)_{\pi_{\lambda,\beta}}^{\kappa}\subset L^{2}(\pi_{\lambda,\beta})\subset(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. The characterization of these spaces is realized via integral transforms and chain of spaces of entire functions of different types and order of growth. Wick calculus extends in a straightforward manner from Gaussian analysis to the present non-Gaussian framework. Finally, in Appendix B we give an explicit relation between (generalized) Appell polynomials and Bell polynomials.