The fractional Poisson measure in infinite dimensions

Abstract

The Mittag-Leffler function $E_α$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_α(ϕ) :=E_α(\int (e^{iϕ(x)}-1)dμ(x)) \] 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 ($α=1$) 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.