  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}.
