Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

How round are the complementary components of planar Brownian motion?

Nina Holden, Şerban Nacu, Yuval Peres, and Thomas S. Salisbury

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Abstract

Consider a Brownian motion $W$ in $\mathbf{C}$ started from $0$ and run for time 1. Let $A(1),A(2),\ldots$ denote the bounded connected components of $\mathbf{C}-W([0,1])$. Let $R(i)$ (resp. $r(i)$) denote the out-radius (resp. in-radius) of $A(i)$ for $i\in\mathbf{N}$. Our main result is that ${\mathbf{E}}[\sum_{i}R(i)^{2}|\log R(i)|^{\theta}]<\infty$ for any $\theta<1$. We also prove that $\sum_{i}r(i)^{2}|\log r(i)|=\infty$ almost surely. These results have the interpretation that most of the components $A(i)$ have a rather regular or round shape.

Résumé

Soit $W$ un mouvement brownien dans $\mathbf{C}$ issu de $0$. Soit $A(1),A(2),\ldots$ les composantes connexes bornées de $\mathbf{C}\setminus W([0,1])$. Soit $R(i)$ (resp. $r(i)$) le rayon extérieur (resp. le rayon intérieur) de $A(i)$, pour $i\in\mathbf{N}$. Notre résultat principal est que $\mathbf{E}[\sum_{i}R(i)^{2}|\log R(i)|^{\theta}]<\infty$ pour tout $\theta<1$. Nous montrons aussi que $\sum_{i}r(i)^{2}|\log r(i)|]=\infty$ presque surement. Ces résultats peuvent s’interpréter comme le fait que la plupart des composantes $A(i)$ ont une forme assez régulière, ou ronde.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 2 (2019), 882-908.

Dates
Received: 24 January 2017
Revised: 24 March 2018
Accepted: 24 March 2018
First available in Project Euclid: 14 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1557820835

Digital Object Identifier
doi:10.1214/18-AIHP902

Subjects
Primary: 60G17: Sample path properties

Keywords
Planar Brownian motion Complementary components of planar Brownian motion

Citation

Holden, Nina; Nacu, Şerban; Peres, Yuval; Salisbury, Thomas S. How round are the complementary components of planar Brownian motion?. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 2, 882--908. doi:10.1214/18-AIHP902. https://projecteuclid.org/euclid.aihp/1557820835


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