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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Variation of multiparameter Brownian motion

Author: Lane Yoder
Journal: Proc. Amer. Math. Soc. 46 (1974), 302-309
MSC: Primary 60J65; Secondary 60G17
MathSciNet review: 0418260
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Abstract: Lévy's $ N$-parameter Brownian motion in $ d$-dimensional space is denoted by $ {W^{(N,d)}}$. Using uniform partitions and a Vitali-type variation, Berman recently extended to $ {W^{(N,1)}}$ a classical result of Lévy concerning the relation between $ {W^{(1,1)}}$ and $ 2$-variation. With this variation $ {W^{(N,d)}}$ has variation dimension $ 2N$ with probability one. An appropriate definition of weak variation is given using powers of the diameters of the images of sets which satisfy a parameter of regularity. A previous result concerning the Hausdorff dimensions of the graph and image is used to show the weak variation dimension of $ {W^{(N,d)}}$ is $ 2N$ with probability one, extending the result for $ {W^{(1,1)}}$ of Goffman and Loughlin. If unrestricted partitions of the domain are used, the weak variation dimension of a function turns out to be the same as the Hausdorff dimension of the image.

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Article copyright: © Copyright 1974 American Mathematical Society