Moments of quantum Lévy areas using sticky shuffle Hopf algebras

  • Robin L. Hudson

    Loughborough University, UK
  • Uwe Schauz

    Xi’an Jiaotong – Liverpool University, Suzhou, China
  • Yue Wu

    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 which deform to the classical Lévy area as . 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.

Cite this article

Robin L. Hudson, Uwe Schauz, Yue Wu, Moments of quantum Lévy areas using sticky shuffle Hopf algebras. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 5 (2018), no. 3, pp. 437–466

DOI 10.4171/AIHPD/59