  Let $W_i=\{W_i(t), t\in \mathbb{R}_+\}, i=1,2$ be two Wiener processes and
$W_3=\{W_3(\mathbf{t}), \mathbf{t}\in \mathbb{R}_+^2\}$ be a two-parameter
Brownian sheet, all three processes being mutually independent. We derive upper
and lower bounds for the boundary non-crossing probability
$$P_f=P\{W_1(t_1)+W_2(t_2)+W_3(\mathbf{t})+h(\mathbf{t})\leq u(\mathbf{t}),
\mathbf{t}\in\mathbb{R}_+^2\},$$ where $h, u: \mathbb{R}_+^2\rightarrow
\mathbb{R}_+$ are two measurable functions. We show further that for large
trend functions $\gamma f>0$ asymptotically when $\gamma \to \infty$ we have
that $\ln P_{\gamma f}$ is the same as $\ln P_{\gamma \underline{f}}$ where
$\underline{f}$ is the projection of $f$ on some closed convex set of the
reproducing kernel Hilbert Space of $W$. It turns out that our approach is
applicable also for the additive Brownian pillow.
