L\'evy area without approximation

TL;DR

This paper introduces a novel, approximation-free definition of the Le9vy area by analyzing the asymptotic behavior of Brownian winding and its average with a Poisson point process, advancing stochastic analysis methods.

Contribution

It provides a new, exact definition of the Le9vy area based on asymptotic estimations, removing the need for path approximations.

Findings

Asymptotic estimations of large Brownian winding areas

Analysis of average winding with Poisson point processes

A new approximation-free definition of the Le9vy area

Abstract

We give asymptotic estimations on the area of the sets of points with large Brownian winding, and study the average winding between a planar Brownian motion and a Poisson point process of large intensity on the plane. This allows us to give a new definition of the L\'evy area which does not rely on approximations of the Brownian path.