The exact Hausdorff measure of the zero set of fractional Brownian motion (Q633137)

A fractional Brownian motion on \(\mathbb R^N\) with index \(H\in (0,1)\) is a zero-mean \(\mathbb R\)-valued Gaussian process \(\{Y(t), t\in \mathbb R^N\}\) with covariance function \(\mathbb E[Y(t)Y(s)]=\frac 12 (|t|^{2H}+|s|^{2H}-|t-s|^{2H})\) and continuous sample paths. Here, \(|\cdot|\) denotes the Euclidean norm. Let \(X\) be a process with values in \(\mathbb R^d\) whose components are independent fractional Brownian motions defined on \(\mathbb R^N\). For \(N>dH\), the authors show that the restriction of the Hausdorff \(\phi\)-measure, where \(\phi(h)=h^{N-dH}(\log\log \frac 1h)^{dH/N}\), to the zero-set \(X^{-1}(0)=\{t\in\mathbb R^N: X(t)=0\}\) coincides, up to a multiplicative constant, with the local time of the process \(X\) at \(0\). This refines a previous result of \textit{Y.\ Xiao} [Probab.\ Theory Relat.\ Fields 109, No.~1, 129--157 (1997; Zbl 0882.60035)].