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Annales de l’Institut Henri Poincaré D

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Volume 5, Issue 3, 2018, pp. 437–466
DOI: 10.4171/AIHPD/59

Published online: 2018-07-25

Moments of quantum Lévy areas using sticky shuffle Hopf algebras

Robin L. Hudson[1], Uwe Schauz[2] and Yue Wu[3]

(1) Loughborough University, UK
(2) Xi’an Jiaotong – Liverpool University, Suzhou, China
(3) Technische Universität Berlin, Germany

We study a family of quantum analogs of Lévy's stochastic area for planar Brownian motion depending on a variance parameter $\sigma \geq 1$ which deform to the classical Lévy area as $\sigma \rightarrow \infty$. 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.

Keywords: Lévy area, non-Fock quantum stochastic calculus, moments, sticky shuffles, Euler numbers

Hudson Robin, Schauz Uwe, Wu Yue: Moments of quantum Lévy areas using sticky shuffle Hopf algebras. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), 437-466. doi: 10.4171/AIHPD/59