Electronic Journal of Probability

Second quantisation for skew convolution products of measures in Banach spaces

David Applebaum and Jan Neerven

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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

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

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

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Zentralblatt MATH identifier

Primary: 81S25: Quantum stochastic calculus
Secondary: 47N30: Applications in probability theory and statistics 60B05: Probability measures on topological spaces 60E07: Infinitely divisible distributions; stable distributions 60G51: Processes with independent increments; Lévy processes 60G57: Random measures 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60H07: Stochastic calculus of variations and the Malliavin calculus

Second quantisation skew convolution family infinitely divisible measure Wiener-Ito decomposition Poisson random measure

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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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