Hausdorff dimensions and Hitting probabilities for some general Gaussian processes

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 $F$.