Hausdorff dimensions and Hitting probabilities for some general Gaussian processes

TL;DR

This paper investigates the Hausdorff dimensions of images of Gaussian processes and their hitting probabilities, providing explicit conditions and bounds based on the variance function and Hausdorff measures.

Contribution

It establishes general conditions on the variance function for the Hausdorff dimension of Gaussian process images to be constant and derives bounds on hitting probabilities in terms of Hausdorff measures.

Findings

Hausdorff dimension of B(E) is constant under certain conditions

Explicit bounds on hitting probabilities involving Hausdorff measure and capacity

Bounds on the Hausdorff dimension of intersections involving B(E) and F

Abstract

Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function $\gamma^2(r)=\operatorname{Var}\left(B_0(r)\right)$ and a canonical metric $\delta(t,s):=(\mathbb{E}\left(B_0(t)-B_0(s)\right)^2)^{1/2}$ which is commensurate with $\gamma(t-s)$. We provide some general condition on $\gamma$ so that for any Borel set $E\subset [0,1]$, the Hausdorff dimension of the image $B(E)$ is constant a.s., and we explicit this constant. Also, we derive under some mild assumptions on $\gamma\,$ an upper and lower bounds of $\mathbb{P}\left\{B(E)\cap F\neq \emptyset \right\}$ in terms of the corresponding Hausdorff measure and capacity of $E\times F$. Some upper and lower bounds for the essential supremum norm of the Hausdorff dimension of $B(E)\cap F$ and $E\cap B^{-1}(F)$ are also given in terms of $d$ and the corresponding Hausdorff dimensions of $E\times F$, $E$, and $F$.