Let
X
(
t
)
X(t)
be a stationary Gaussian process,
f
(
t
)
f(t)
a continuous function, and
T
T
a finite or infinite interval. This paper develops asymptotic estimates for
P
(
X
(
t
)
⩾
f
(
t
)
P(X(t) \geqslant f(t)
, some
t
∈
T
t \in T
when this probability is small. After transformation to an Ornstein Uhlenbeck process the results are also applicable to Brownian motion. In that special case, if
W
(
t
)
W(t)
is Brownian motion,
f
f
is continuously differentiable, and
T
=
[
0
,
T
]
T = [0,T]
our estimate for
P
(
W
(
t
)
⩾
f
(
t
)
P(W(t) \geqslant f(t)
, some
t
∈
T
)
t \in T)
is
\[
Λ
=
∫
0
T
(
2
t
)
−
1
(
f
(
t
)
/
t
1
/
2
)
ϕ
(
f
(
t
)
/
t
1
/
2
)
d
t
+
I
{
(
f
(
t
)
/
t
1
/
2
)
′
|
t
=
T
>
0
}
Φ
∗
(
f
(
T
)
/
T
1
/
2
)
\Lambda = \int _0^T {{{(2t)}^{ - 1}}(f(t)/{t^{1/2}})\phi (f(t)/{t^{1/2}})} dt + {I_{\{ (f(t)/{t^{1/2}})’{|_{t = T}} > 0\} }}{\Phi ^ \ast }(f(T)/{T^{1/2}})
\]
provided
Λ
\Lambda
is small. Here
ϕ
\phi
is the standard normal density and
Φ
∗
{\Phi ^ \ast }
is its upper tail distribution. Our approach is to find an approximate first passage density and then compute crossing probabilities as a one-dimensional integral. In the case of boundaries without cusps, our results unify and extend separate results for crossings of constant levels developed by Pickands, and Qualls-Watanabe, and crossings of rapidly increasing barriers studied by Berman. Applications are also briefly explored.