  This paper considers extreme values attained by a centered, multidimensional
Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift
$d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity
conditions, we establish the asymptotics of \[\log\mathbb P\left(\exists{t\in
T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right),\] for positive
thresholds $q_i>0$, $i=1,\ldots,n$, and $u\to\infty$. Our findings generalize
and extend previously known results for the single-dimensional and
two-dimensional cases. A number of examples illustrate the theory.
