Irregularity scales for Gaussian processes: Hausdorff dimensions and hitting probabilities

TL;DR

This paper investigates the Hausdorff dimensions of images and graphs of Gaussian processes with irregular variance functions, establishing their almost sure constancy and providing explicit formulas and bounds for hitting probabilities.

Contribution

It introduces a new method using Karhunen-Loève expansion to prove the almost sure constancy of Hausdorff dimensions even for highly irregular Gaussian processes.

Findings

Hausdorff dimensions are almost surely constant under mild regularity conditions.

Explicit formulas relate dimensions to the process's metric and set dimensions.

Bounds on hitting probabilities depend on Hausdorff measure and capacity in a specific metric.

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.