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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}$\end{document} , and mφ(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.