On note un mouvement brownien linéaire et dim la dimension de Hausdorff. Pour et , nous montrons que, presque sûrement, il n'existe pas d'ensemble tel que et 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 tel que et 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 be a linear Brownian motion and let dim denote the Hausdorff dimension. Let and . We prove that, almost surely, there exists no set such that and 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 such that and has finite β-variation. The zero set of B and a deterministic construction witness that the above theorems give the optimal dimensions.
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Richárd Balka 1, 2 ; Yuval Peres 3
@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}, }
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/
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