Irregularity scales for Gaussian processes: Hausdorff dimensions and hitting probabilities

Abstract

Let $X$ be a $d$-dimensional Gaussian process in $[0,1]$, where the component are independent copies of a scalar Gaussian process $X_0$ on $[0,1]$ with a given general variance function $\gamma^2(r)=\operatorname{Var}\left(X_0(r)\right)$ and a canonical metric $\delta(t,s):=(\mathbb{E}\left(X_0(t)-X_0(s)\right)^2)^{1/2}$ which is commensurate with $\gamma(t-s)$. Under a weak regularity condition on $\gamma$, referred to below as $\mathbf{(C_{0+})}$, which allows $\gamma$ to be far from H\"older-continuous, we prove that for any Borel set $E\subset [0,1]$, the Hausdorff dimension of the image $X(E)$ and of the graph $Gr_E(X)$ are constant almost surely. Furthermore, we show that these constants can be explicitly expressed in terms of $\dim_{\delta}(E)$ and $d$. However, when $\mathbf{(C_{0+})}$ is not satisfied, the classical methods may yield different upper and lower bounds for the underlying Hausdorff dimensions. This case is illustrated via a class of highly irregular processes known as logBm. Even in such cases, we employ a new method to establish that the Hausdorff dimensions of $X(E)$ and $Gr_E(X)$ are almost surely constant. The method uses the Karhunen-Lo\`eve expansion of $X$ to prove that these Hausdorff dimensions are measurable with respect to the expansion's tail sigma-field. Under similarly mild conditions on $\gamma$, we derive upper and lower bounds on the probability that the process $X$ can reach the Borel set $F$ in $\mathbb{R}^d$ from the Borel set $E$ in $[0,1]$. These bounds are obtained by considering the Hausdorff measure and the Bessel-Riesz capacity of $E\times F$ in an appropriate metric $\rho_{\delta}$ on the product space, relative to appropriate orders. Moreover, we demonstrate that the dimension $d$ plays a critical role in determining whether $X\lvert_E$ hits $F$ or not.