Hitting probabilities for general Gaussian processes

Abstract

For a scalar Gaussian process $B$ on $\mathbb{R}_{+}$ with a prescribed general variance function $\gamma^{2}\left(r\right) =\mathrm{Var}\left(B\left(r\right) \right) $ and a canonical metric $\mathrm{E}[\left(B\left(t\right) -B\left(s\right) \right) ^{2}]$ which is commensurate with $\gamma^{2}\left(t-s\right) $, we estimate the probability for a vector of $d$ iid copies of $B$ to hit a bounded set $A$ in $\mathbb{R}^{d}$, with conditions on $\gamma$ which place no restrictions of power type or of approximate self-similarity, assuming only that $\gamma$ is continuous, increasing, and concave, with $\gamma\left(0\right) =0$ and $\gamma^{\prime}\left(0+\right) =+\infty$. We identify optimal base (kernel) functions which depend explicitly on $\gamma$, to derive upper and lower bounds on the hitting probability in terms of the corresponding generalized Hausdorff measure and non-Newtonian capacity of $A$ respectively. The proofs borrow and extend some recent progress for hitting probabilities estimation, including the notion of two-point local-nondeterminism in Bierm\'{e}, Lacaux, and Xiao \cite{Bierme:09}.