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

@article{CRMATH_2014__352_12_1057_0,
     author = {Balka, Rich\'ard and Peres, Yuval},
     title = {Restrictions of {Brownian} motion},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1057--1061},
     publisher = {Elsevier},
     volume = {352},
     number = {12},
     year = {2014},
     doi = {10.1016/j.crma.2014.09.023},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2014.09.023/}
}

TY  - JOUR
AU  - Balka, Richárd
AU  - Peres, Yuval
TI  - Restrictions of Brownian motion
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 1057
EP  - 1061
VL  - 352
IS  - 12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2014.09.023/
DO  - 10.1016/j.crma.2014.09.023
LA  - en
ID  - CRMATH_2014__352_12_1057_0
ER  -

%0 Journal Article
%A Balka, Richárd
%A Peres, Yuval
%T Restrictions of Brownian motion
%J Comptes Rendus. Mathématique
%D 2014
%P 1057-1061
%V 352
%N 12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2014.09.023/
%R 10.1016/j.crma.2014.09.023
%G en
%F CRMATH_2014__352_12_1057_0

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. http://www.numdam.org/articles/10.1016/j.crma.2014.09.023/

Bibliographie
Cité par

[1] Angel, O.; Balka, R.; Peres, Y. Increasing subsequences of random walks (preprint) | arXiv

[2] Antunović, T.; Burdzy, K.; Peres, Y.; Ruscher, J. Isolated zeros for Brownian motion with variable drift, Electron. J. Probab., Volume 16 (2011) no. 65, pp. 1793-1814

[3] Elekes, M. Hausdorff measures of different dimensions are isomorphic under the Continuum Hypothesis, Real Anal. Exch., Volume 30 (2004) no. 2, pp. 605-616

[4] Falconer, K. Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, 2003

[5] Hawkes, J. On the Hausdorff dimension of the intersection of the range of a stable process with a Borel set, Z. Wahrscheinlichkeit, Volume 19 (1971), pp. 90-102

[6] Kahane, J.-P.; Katznelson, Y. Restrictions of continuous functions, Isr. J. Math., Volume 174 (2009), pp. 269-284

[7] Kaufman, R. Une propriété métrique du mouvement brownien, C. R. Acad. Sci. Paris, Volume 268 (1969), pp. 727-728

[8] Kaufman, R. Measures of Hausdorff-type, and Brownian motion, Mathematika, Volume 19 (1972), pp. 115-119

[9] Lévy, P. Théorie de l'addition des variables aléatoires, Gauthier–Villars, Paris, 1937

[10] Máthé, A. Measurable functions are of bounded variation on a set of Hausdorff dimension $\frac{1}{2}$ , Bull. Lond. Math. Soc., Volume 45 (2013), pp. 580-594

[11] Mattila, P. Geometry of Sets and Measures in Euclidean Spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, UK, 1995

[12] Mörters, P.; Peres, Y. Brownian Motion, with an appendix by Oded Schramm and Wendelin Werner, Cambridge University Press, Cambridge, UK, 2010

Cité par Sources :