Brownian Motion and Dimension of Perfect Sets

Kaufman, Robert

Canadian journal of mathematics, Tome 22 (1970) no. 3, pp. 674-680

Résumé

Let X(t) denote real-valued Brownian motion on the interval 0 ≦ t ≦ 1, so normalized that E(X2(t)) = t. We prove some theorems about transforms X(F) of closed sets F: in general, F is not known in advance but depends on X. The main point of comparison among sets is taken to be their Hausdorff dimension, and in this respect the linear process is quite different from the planar. We state and discuss briefly two theorems.(A) It is almost sure that, for every closed set F in [0, 1], (B) For each closed set F in (—00,00) and number a , Plainly, statements (A) and (B) are nearly best possible. For the planar process dimX(F) = 2 dim F (with the same quantification as in (A)) [6].

Détail

DOI : 10.4153/CJM-1970-074-3

Bibliographie

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