Restrictions of Brownian motion

Balka, Richárd; Peres, Yuval

doi:10.1016/j.crma.2014.09.023

Probability theory

Restrictions of Brownian motion
[Restrictions du mouvement brownien]

Présenté par : Jean-François Le Gall

Balka, Richárd ^1,² ; Peres, Yuval ³

¹ Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA
² Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, PO Box 127, 1364 Budapest, Hungary
³ Microsoft Research, 1 Microsoft Way, Redmond, WA 98052, USA

Comptes Rendus. Mathématique, Tome 352 (2014) no. 12, pp. 1057-1061.

Résumé
Abstract

On note ${B (t) : 0 \leq t \leq 1}$ un mouvement brownien linéaire et dim la dimension de Hausdorff. Pour $α > \frac{1}{2}$ et $1 \leq β \leq 2$ , nous montrons que, presque sûrement, il n'existe pas d'ensemble $A \subset [0, 1]$ tel que $\dim A > \frac{1}{2}$ et $B : A \to R$ soit α-Hölder continue. La preuve est une application du théorème de Kaufman sur le doublement de dimension. Comme corollaire du théorème ci-dessus, nous montrons que, presque sûrement, il n'existe pas d'ensemble $A \subset [0, 1]$ tel que $\dim A > \frac{β}{2}$ et $B : A \to R$ ait une β-variation finie. L'ensemble des zéros de B et une construction déterministe montrent que les théorèmes ci-dessus donnent les dimensions optimales.

Let ${B (t) : 0 \leq t \leq 1}$ be a linear Brownian motion and let dim denote the Hausdorff dimension. Let $α > \frac{1}{2}$ and $1 \leq β \leq 2$ . We prove that, almost surely, there exists no set $A \subset [0, 1]$ such that $\dim A > \frac{1}{2}$ and $B : A \to R$ is α-Hölder continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A \subset [0, 1]$ such that $\dim A > \frac{β}{2}$ and $B : A \to R$ has finite β-variation. The zero set of B and a deterministic construction witness that the above theorems give the optimal dimensions.

Reçu le : 2014-06-26
Accepté le : 2014-09-26
Publié le : 2014-10-16

DOI : 10.1016/j.crma.2014.09.023

Affiliations des auteurs :

Balka, Richárd ^{1, 2} ; Peres, Yuval ³

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     title = {Restrictions of {Brownian} motion},
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     pages = {1057--1061},
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Balka, Richárd; Peres, Yuval. Restrictions of Brownian motion. Comptes Rendus. Mathématique, Tome 352 (2014) no. 12, pp. 1057-1061. doi : 10.1016/j.crma.2014.09.023. https://www.numdam.org/articles/10.1016/j.crma.2014.09.023/

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