A biorthogonal approach to the infinite dimensional fractional Poisson measure

Abstract

In this paper we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure ${\pi }_{\sigma }^{\beta }$, $0<\beta \le 1$, on the dual of Schwartz test function space ${\mathcal{D}}^{\prime }$. The Hilbert space $L^2({\pi }_{\sigma }^{\beta })$ of complex-valued functions is described in terms of a system of generalized Appell polynomials ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$ associated to the measure ${\pi }_{\sigma }^{\beta }$. The kernels $C_n^{\sigma ,\beta }(\cdot )$, $n\in {\mathbb{N}}_0$, of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$, there is a generalized dual Appell system ${\mathbb{Q}}^{\sigma ,\beta ,\alpha }$ that is biorthogonal to ${\mathbb{P}}^{\sigma ,\beta ,\alpha }$. The test and generalized function spaces associated to the measure ${\pi }_{\sigma }^{\beta }$ are completely characterized using an integral transform as entire functions.

39 pages, 1 figure

References in corpus (2)

• Fractional processes: from Poisson to branching one
• Stirling operators in spatial combinatorics