Hausdorff measure of trajectories of multiparameter fractional Brownian motion
For \(0 < \alpha < 1\), let \(Y = \{Y(t) : t \in \mathbb{R}^N\}\) be the real- valued Gaussian process with covariance \(E(Y(t) Y(s)) = |t |^{2 \alpha} + |s |^{2 \alpha} - |t - s |^{2 \alpha}\), where \(|\cdot |\) denotes the \(N\)-dimensional Euclidean norm, \(N \geq 1\). Let \(X^1, \ldots, X^d\) be independent copies of \(Y\), and consider the \(\mathbb{R}^d\)-valued process \(\{X(t) = (X^1(t), \ldots, X^d(t)) : t \in \mathbb{R}^N\}\). It is shown that if \(N < \alpha d\), then a.s. for any compact subset \(L\) of \(\mathbb{R}^N\) with nonempty interior one has \(0 < \mu_\varphi (X(L)) < \infty\), where \(\mu_\varphi\) denotes the Hausdorff measure associated with the function \(\varphi (\varepsilon) = \varepsilon^{N/ \alpha} \log \log (1/ \varepsilon)\). This result extends the work of \textit{A. Goldman} [``Mouvement brownien à plusieurs paramètres: mesure de Hausdorff des trajectoires'' (1988; Zbl 0681.60040)] in the case \(\alpha = 1/2\). At the same time, the present author's proof is considerably simpler.
scientific article; zbMATH DE number 796519