  We study boundary non-crossing probabilities $$ P_{f,u} := \mathrm
P\big(\forall t\in \mathbb T\ X_t + f(t)\le u(t)\big) $$ for continuous
centered Gaussian process $X$ indexed by some arbitrary compact separable
metric space $\mathbb T$. We obtain both upper and lower bounds for $P_{f,u}$.
The bounds are matching in the sense that they lead to precise logarithmic
asymptotics for the large-drift case $P_{y f,u}$, $y \to+\infty$, which are
two-term approximations (up to $o(y)$). The asymptotics are formulated in terms
of the solution $\tilde f$ to the constrained optimization problem $$
\|h\|_{\mathbb H_X}\to \min, \quad h\in \mathbb H_X, h\ge f $$ in the
reproducing kernel Hilbert space $\mathbb H_X$ of $X$. Several applications of
the results are further presented.
