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

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Abstract

Let {X(t), t∈ℝN} be a fractional Brownian motion in ℝd of index H. If L(0,I) is the local time of X at 0 on the interval I⊂ℝN, 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 ℝN.

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  1. Département de Mathématiques, École Polytechnique Fédérale, 1015, Lausanne, Switzerland

    D. Baraka & T. S. Mountford

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  1. D. Baraka
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  2. T. S. Mountford
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Correspondence to T. S. Mountford.

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Baraka, D., Mountford, T.S. The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion. J Theor Probab 24, 271–293 (2011). https://doi.org/10.1007/s10959-009-0271-1

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