Variation of multiparameter Brownian motion

Yoder, Lane

doi:10.1090/S0002-9939-1974-0418260-4

Variation of multiparameter Brownian motion
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Proc. Amer. Math. Soc. 46 (1974), 302-309 Request permission

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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