Hausdorff Measure of Trajectories of Multiparameter Fractional Brownian Motion

Abstract

Consider $0 < \alpha < 1$ and the Gaussian process $Y(t)$ on $\mathbb{R}^N$ with covariance $E(Y(t)Y(s)) = |t|^{2\alpha} + |s|^{2\alpha} - |t - s|^{2\alpha}$, where $|t|$ is the Euclidean norm of $t$. Consider independent copies $X^1,\ldots,X^d$ of $Y$ and the process $X(t) = (X^1(t),\ldots,X^d(t))$ valued in $\mathbb{R}^d$. In the transient case $(N < \alpha d)$ we show that a.s. for each compact set $L$ of $\mathbb{R}^N$ with nonempty interior, we have $0 < \mu_\varphi(X(L)) < \infty$, where $\mu_\varphi$ denotes the Hausdorff measure associated with the function $\varphi(\varepsilon) = \varepsilon^{N/\alpha} \log \log(1/\varepsilon)$. This result extends work of A. Goldman in the case $\alpha = 1/2$; the proofs are considerably simpler.