The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion

D. Baraka and T. S. Mountford ()
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D. Baraka: École Polytechnique Fédérale
T. S. Mountford: École Polytechnique Fédérale

Journal of Theoretical Probability, 2011, vol. 24, issue 1, 271-293

Abstract: Abstract Let {X(t), t∈ℝ N } be a fractional Brownian motion in ℝ d of index H. If L(0,I) is the local time of X at 0 on the interval I⊂ℝ N , then there exists a positive finite constant c(=c(N,d,H)) such that $$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$$ where $\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}$ , and m φ (E) is the Hausdorff φ-measure of E. This refines a previous result of Xiao (Probab. Theory Relat. Fields 109: 126–197, 1997) on the relationship between the local time and the Hausdorff measure of zero set for d-dimensional fractional Brownian motion on ℝ N .

Keywords: Local times; Hausdorff measures; Level sets; Fractional Brownian motion; 60G60; 60G15; 60G17 (search for similar items in EconPapers)
Date: 2011
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DOI: 10.1007/s10959-009-0271-1

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