Probability theory
Restrictions of Brownian motion
Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 1057-1061.

Let ${B(t):0≤t≤1}$ be a linear Brownian motion and let dim denote the Hausdorff dimension. Let $α>12$ and $1≤β≤2$. We prove that, almost surely, there exists no set $A⊂[0,1]$ such that $dim⁡A>12$ and $B:A→R$ 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 $A⊂[0,1]$ such that $dim⁡A>β2$ and $B:A→R$ has finite β-variation. The zero set of B and a deterministic construction witness that the above theorems give the optimal dimensions.

On note ${B(t):0≤t≤1}$ un mouvement brownien linéaire et dim la dimension de Hausdorff. Pour $α>12$ et $1≤β≤2$, nous montrons que, presque sûrement, il n'existe pas d'ensemble $A⊂[0,1]$ tel que $dim⁡A>12$ et $B:A→R$ 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 $A⊂[0,1]$ tel que $dim⁡A>β2$ et $B:A→R$ 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.

Accepted:
Published online:
DOI: 10.1016/j.crma.2014.09.023

Richárd Balka 1, 2; Yuval Peres 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
@article{CRMATH_2014__352_12_1057_0,
author = {Rich\'ard Balka and Yuval Peres},
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},
}
TY  - JOUR
AU  - Richárd Balka
AU  - Yuval Peres
TI  - Restrictions of Brownian motion
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 1057
EP  - 1061
VL  - 352
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2014.09.023
LA  - en
ID  - CRMATH_2014__352_12_1057_0
ER  - 
%0 Journal Article
%A Richárd Balka
%A Yuval Peres
%T Restrictions of Brownian motion
%J Comptes Rendus. Mathématique
%D 2014
%P 1057-1061
%V 352
%N 12
%I Elsevier
%R 10.1016/j.crma.2014.09.023
%G en
%F CRMATH_2014__352_12_1057_0
Richárd Balka; Yuval Peres. Restrictions of Brownian motion. Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 1057-1061. doi : 10.1016/j.crma.2014.09.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.09.023/

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

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

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

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

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

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

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

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

[10] A. Máthé Measurable functions are of bounded variation on a set of Hausdorff dimension $12$, Bull. Lond. Math. Soc., Volume 45 (2013), pp. 580-594

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

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

Cited by Sources: