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Hausdorff measure of the graph of fractional Brownian motion

Published online by Cambridge University Press:  01 November 1997

YIMIN XIAO
YIMIN XIAO
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, U.S.A. e-mail: xiao@math.utah.edu
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Abstract

Let X(t) (t ∈ RN) be a fractional Brownian motion in Rd of index α. Let GrX([0, 1]N) be the graph of X and let

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and

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It is proved that if N<αd, then almost surely

formula here

and if N>αd, then almost surely

formula here

where ϕ-m is the ϕ-Hausdorff measure and K1, K2 are positive finite constants. The exact Hausdorff measure of the image and graph of certain Gaussian random fields with independent fractional Brownian motion components are also obtained.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 122 , Issue 3 , November 1997 , pp. 565 - 576
Copyright
Cambridge Philosophical Society 1997

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