Analysis on Poisson and Gamma Spaces

Abstract

We study the spaces of Poisson, compound Poisson and Gamma noises as special cases of a general approach to non-Gaussian white noise calculus, see Ref. 18. We use a known unitary isomorphism between Poisson and compound Poisson spaces in order to transport analytic structures from Poisson space to compound Poisson space. Finally we study a Fock type structure of chaos decomposition on Gamma space.

The Square of White Noise as a Jacobi Field

“…On the other hand, in papers [16,19,20,11] (see also [17,12,10,13]), the Jacobi field of the Lévy processes of Meixner's type, i.e., the gamma, Pascal, and Meixner processes, was studied. Let us shortly explain this approach.…”

WICK-TYPE STOCHASTIC KdV EQUATION BASED ON LÉVY WHITE NOISE

“…Furthermore, Chen and Xie [3][4][5][6], Xie [7][8][9], Ghany [10][11][12], Ghany et al [13], and Ghany and Hyder [14][15][16][17][18] investigated some stochastic travelling wave equations using Gaussian white noise analysis. On the other hand, an extension of Gaussian white noise analysis to non-Gaussian white noise analysis was established in [21], and developed further in [22,23]. Based on this extension, Løkka et al [24] and Øksendal [25] developed a white noise framework for the study of SPDEs driven by a d-parameter Lévy white noise, which is in fact a non-Gaussian white noise.…”

Lévy White Noise Calculus Based on Interaction Exponents

“…Since then the Gaussian white noise analysis has been fully developed with considerable applications in stochastic analysis, mathematical finance and mathematical physics. Recently it has been greatly developed in non-Gaussian cases (e.g., the Poisson white noise analysis in [2,6,10,12,17,26], the gamma white noise calculus in [3,11,12,18], the Pascal and Meixner white noise calculus in [18,19] and general cases in [1,2,7,8,13,15,22,23,27]), especially in the chaotic representation of Lévy white noise functionals. Basically there are two strategies in these constructions: the first one uses vector-valued (i.e., L 2 (n)-valued, where n is the Lévy measure) Fock expansion in terms of iterated integrals with respect to the compensated Poisson random measure (cf.…”