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May 2019 How round are the complementary components of planar Brownian motion?
Nina Holden, Şerban Nacu, Yuval Peres, Thomas S. Salisbury
Ann. Inst. H. Poincaré Probab. Statist. 55(2): 882-908 (May 2019). DOI: 10.1214/18-AIHP902

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.

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.

Citation

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Nina Holden. Şerban Nacu. Yuval Peres. Thomas S. Salisbury. "How round are the complementary components of planar Brownian motion?." Ann. Inst. H. Poincaré Probab. Statist. 55 (2) 882 - 908, May 2019. https://doi.org/10.1214/18-AIHP902

Information

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

zbMATH: 07097335
MathSciNet: MR3949957
Digital Object Identifier: 10.1214/18-AIHP902

Subjects:
Primary: 60G17

Rights: Copyright © 2019 Institut Henri Poincaré

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Vol.55 • No. 2 • May 2019
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