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November 2022 Lévy area without approximation
Isao Sauzedde
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 58(4): 2165-2200 (November 2022). DOI: 10.1214/21-AIHP1230


We give asymptotic estimations on the area of the set 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 Lévy area which does not rely on piecewise linear approximations of the Brownian path.

On donne une estimation asymptotique de l’aire de l’ensemble des points autour desquels le mouvement brownian s’enlace un grand nombre de fois, et on étudie l’enlacement moyen entre un mouvement brownian plan et un processus de Poisson de grande intensité dans le plan. Cela nous permet de donner une nouvelle définition de l’aire de Lévy, qui ne repose pas sur des approximations linéaires par morceaux du chemin brownien.

Funding Statement

I am pleased to acknowledge support from the École Normale Supérieure de Lyon, the Laboratoire de Probabilités, Statistique et Modélisation and Sorbonne Université when I was writing this paper, as well as from the ERC Advanced Grant 740900 (LogCorRM) during the editorial process.


I would like to thank my former PhD advisor Thierry Lévy for his continual help, Pierre Perruchaud for the fruitful discussions concerning this work, and the referee for his careful reading. I am also thankful to Jean-François Le Gall for a discussion that helped me realise that the result of Theorem 1 was implicit in a previous version of the paper.


Download Citation

Isao Sauzedde. "Lévy area without approximation." Ann. Inst. H. Poincaré Probab. Statist. 58 (4) 2165 - 2200, November 2022.


Received: 22 January 2021; Revised: 14 October 2021; Accepted: 9 November 2021; Published: November 2022
First available in Project Euclid: 6 October 2022

Digital Object Identifier: 10.1214/21-AIHP1230

Primary: 60J65
Secondary: 60B20 , 60E07

Keywords: Lévy’s area , Planar Brownian motion , Stokes’ formula

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré


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Vol.58 • No. 4 • November 2022
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