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

Author

Listed:
  • D. Baraka

    (École Polytechnique Fédérale)

  • T. S. Mountford

    (École Polytechnique Fédérale)

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 .

Suggested Citation

  • D. Baraka & T. S. Mountford, 2011. "The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion," Journal of Theoretical Probability, Springer, vol. 24(1), pages 271-293, March.
    • Handle: RePEc:spr:jotpro:v:24:y:2011:i:1:d:10.1007_s10959-009-0271-1
    DOI: 10.1007/s10959-009-0271-1
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    References listed on IDEAS

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    1. Berman, Simeon M., 1987. "Spectral conditions for local nondeterminism," Stochastic Processes and their Applications, Elsevier, vol. 27, pages 73-84.
    2. Kasahara, Y. & Kôno, N. & Ogawa, T., 1999. "On tail probability of local times of Gaussian processes," Stochastic Processes and their Applications, Elsevier, vol. 82(1), pages 15-21, July.
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