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Moments of quantum Lévy areas using sticky shuffle Hopf algebras

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posted on 17.09.2018, 11:12 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 Interactions

Volume

5

Issue

3

Pages

437 - 466

Citation

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 Society

Version

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

10/06/2017

Publication date

2018

Notes

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/59

ISSN

2308-5827

eISSN

2308-5835

Language

en

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