Moments of quantum L\'evy areas using sticky shuffle Hopf algebras

TL;DR

This paper investigates quantum analogs of Lévý's stochastic area for planar Brownian motion, utilizing sticky shuffle Hopf algebras to compute moments and analyze their deformation to classical values as variance increases.

Contribution

It introduces a new quantum deformation framework for Lévý areas using Hopf algebra structures and computes their moments explicitly.

Findings

Quantum Lévý areas depend on a variance parameter and deform to classical areas as variance increases.

Moments of quantum Lévý areas are evaluated using Hopf algebra techniques.

Classical moments are related to Euler numbers, recovered in the infinite variance limit.

Abstract

We study a family of quantum analogs of L\'evy's stochastic area for planar Brownian motion depending on a variance parameter $\sigma \geq 1$ which deform to the classical L\'evy 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 Heisnberg-type commutation relations. Such iterated integrals can be multiplied using the sticky shuffle product determined by the underlying It\^o algebra of stochastic differentials. We use the corresponding Hopf algebra structure to evaluate the moments of the quantum L\'evy 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.