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Brownian Motion and Dimension of Perfect Sets

Published online by Cambridge University Press:  20 November 2018

Robert Kaufman
Robert Kaufman*
Affiliation:
University of Illinois, Urbana, Illinois
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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].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 3 , 01 June 1970 , pp. 674 - 680
Copyright
Copyright © Canadian Mathematical Society 1970

References

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