Summary
For each −∞≦c 1<c 2≦∞, consider the set of one-sided slow points of the Brownian path, B, defined by
. We find the Hausdorff dimension of S +(c 1, c 2)(ω) as well as S +(c 1, c 2)∩A(ω), where A a progressively measurable set satisfying certain hypotheses. In particular, the Hausdorff dimension of the two-sided slow points, S(c 1, c 2), and the (one-sided and two-sided) slow points in the zero set of B are obtained by making an appropriate choice of A. These results show, for example, that S(−c,c)≠Ø or = Ø, according as c is greater than or less than the smallest positive zero of M(−1/2, 1/2, x 2/2) (≅ 1.3069), where M is the confluent hypergeometric function. This confirms a conjecture of Burgess Davis, whose methods play a major role in the proofs. The results are also extended to higher dimensions.
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Perkins, E. On the Hausdorff dimension of the Brownian slow points. Z. Wahrscheinlichkeitstheorie verw Gebiete 64, 369–399 (1983). https://doi.org/10.1007/BF00532968
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DOI: https://doi.org/10.1007/BF00532968