Probabilistic Interpretation of Number Operator Acting on Bernoulli Functionals

Jing Zhang, Lixia Zhang and Caishi Wang
Additional contact information
Jing Zhang: School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Lixia Zhang: School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China
Caishi Wang: School of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, China

Mathematics, 2022, vol. 10, issue 15, 1-11

Abstract: Let N be the number operator in the space H of real-valued square-integrable Bernoulli functionals. In this paper, we further pursue properties of N from a probabilistic perspective. We first construct a nuclear space G , which is also a dense linear subspace of H , and then by taking its dual G * , we obtain a real Gel’fand triple G ⊂ H ⊂ G * . Using the well-known Minlos theorem, we prove that there exists a unique Gauss measure γ N on G * such that its covariance operator coincides with N . We examine the properties of γ N , and, among others, we show that γ N can be represented as a convolution of a sequence of Borel probability measures on G * . Some other results are also obtained.

Keywords: Bernoulli functionals; number operator; Gel’fand triple; Gauss measure; convolution of measures (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/10/15/2635/pdf (application/pdf)
https://www.mdpi.com/2227-7390/10/15/2635/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:10:y:2022:i:15:p:2635-:d:873249

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().