## Electronic Journal of Probability

### Second quantisation for skew convolution products of measures in Banach spaces

#### Abstract

We study measures in Banach space which arise as the skew convolution product of two other measures where the convolution is deformed by a skew map. This is the structure that underlies both the theory of Mehler semigroups and operator self-decomposable measures. We show how that given such a set-up the skew map can be lifted to an operator that acts at the level of function spaces and demonstrate that this is an example of the well known functorial procedure of second quantisation. We give particular emphasis to the case where the product measure is infinitely divisible and study the second quantisation process in some detail using chaos expansions when this is either Gaussian or is generated by a Poisson random measure.

#### Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 11, 17 pp.

Dates
Accepted: 17 January 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065653

Digital Object Identifier
doi:10.1214/EJP.v19-3031

Mathematical Reviews number (MathSciNet)
MR3164764

Zentralblatt MATH identifier
1291.60012

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

#### Citation

Applebaum, David; Neerven, Jan. Second quantisation for skew convolution products of measures in Banach spaces. Electron. J. Probab. 19 (2014), paper no. 11, 17 pp. doi:10.1214/EJP.v19-3031. https://projecteuclid.org/euclid.ejp/1465065653

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