Pascal white noise calculus

Abstract

In this paper white noise analysis with respect to the Lévy process with negative binomial distributed marginals is investigated. An appropriate space of distributions, ℰ ′, is used to describe the structure of the Hilbert space of quadratic integrable functionals with respect to the Pascal white noise measure Λ_NB. The constructed decomposition is used to define a nuclear triple of test and generalized functions, where θ is a Young function satisfying some suitable conditions. By using the 𝒮-transform and the symbol transform σ_NB, a general characterization theorems are proven for Pascal white noise distributions, white noise test functions and white noise operators in terms of analytical functions with growth condition of exponential type. As application, some quantum stochastic differential equations are solved with special emphasis on Wick calculus.

Keywords:

• Pascal white noise
• polynomial functionals
• 𝒮-transform
• symbol transform
• quantum stochastic differential equation
• Wick calculus

AMS Subject Classification:

• Primary: 60H40
• Secondary: 46A32, 46F25, 46G20

Notes

^1. [email protected].

^2. [email protected].

Additional information

Notes on contributors

Habib Ouerdiane

1. 1. [email protected].

Anis Riahi

2. 2. [email protected].