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Moments of quantum Lévy areas using sticky shuffle Hopf algebras
journal contribution
posted on 2018-09-17, 11:12 authored by Robin Hudson, Uwe Schauz, Yue Wu© European Mathematical Society. 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.
History
School
- Science
Department
- Mathematical Sciences
Published in
Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their InteractionsVolume
5Issue
3Pages
437 - 466Citation
HUDSON, R.L., SCHAUZ, U. and WU, Y., 2018. Moments of quantum Lévy areas using sticky shuffle Hopf algebras. Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions, 5(3), pp. 437-466.Publisher
© European Mathematical SocietyVersion
- AM (Accepted Manuscript)
Publisher statement
This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/Acceptance date
2017-06-10Publication date
2018Notes
This paper was accepted for publication in the journal Annales de l'Institut Henri Poincare (D) Combinatorics, Physics and their Interactions and the definitive published version is available at https://doi.org/10.4171/AIHPD/59ISSN
2308-5827eISSN
2308-5835Publisher version
Language
- en