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Main Title: Moments of quantum Lévy areas using sticky shuffle Hopf algebras
Author(s): Hudson, Robin
Schauz, Uwe
Wu, Yue
Type: Article
Language Code: en
Abstract: We study a family of quantum analogs of Lévy's stochastic area for planar Brownian motion depending on a variance parameter σ ≥ 1 which deform to the classical Lévy area as σ → ∞. They are defined as second rank iterated stochastic integrals against the components of planar Brownian motion, which are one-dimensional Brownian motions satisfying Heisenberg-type commutation relations. Such iterated integrals can be multiplied using the sticky shuffle product determined by the underlying Itô algebra of stochastic differentials. We use the corresponding Hopf algebra structure to evaluate the moments of the quantum Lévy areas and study how they deform to their classical values, which are well known to be given essentially by the Euler numbers, in the infinite variance limit.
Issue Date: 2018
Date Available: 30-Jan-2019
DDC Class: 510 Mathematik
530 Physik
Subject(s): Lévy area
non-Fock quantum stochastic calculus
sticky shuffles
Euler numbers
Journal Title: Annales de l'Institut Henri Poincaré D
Publisher: European Mathematical Society
Publisher Place: Zürich
Volume: 5
Issue: 3
Publisher DOI: 10.4171/AIHPD/59
Page Start: 437
Page End: 466
EISSN: 2308-5835
ISSN: 2308-5827
Notes: Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.
This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.
Appears in Collections:Inst. Mathematik » Publications

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