The fractional Poisson measure in infinite dimensions

TL;DR

This paper introduces a new infinite-dimensional fractional Poisson measure based on the Mittag-Leffler function, extending Poisson analysis with explicit operator formulas and notable differences from classical Poisson measures.

Contribution

It constructs a fractional Poisson measure in infinite dimensions, proving its properties and developing a related analysis framework based on configuration space harmonic analysis.

Findings

The fractional Poisson measure's characteristic functional satisfies the Bochner-Minlos theorem.

Explicit formulas for annihilation, creation, and second quantization operators are derived.

The measure's support coincides with that of the classical Poisson measure, despite notable differences.

Abstract

The Mittag-Leffler function $E_{\alpha}$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_{\alpha}(\phi) :=E_{\alpha}(\int (e^{i\phi(x)}-1)d\mu (x)) \] which we prove to fulfill all requirements of the Bochner-Minlos theorem. The identity of the support of this new measure with the support of the infinite-dimensional Poisson measure ($\alpha =1$) allows the development of a fractional infinite-dimensional analysis modeled on Poisson analysis through the combinatorial harmonic analysis on configuration spaces. This setting provides, in particular, explicit formulas for annihilation, creation, and second quantization operators. In spite of the identity of the supports, the fractional Poisson measure displays some noticeable differences in relation to the Poisson measure, which may be physically quite significant.