Consider the first exit time $\tau_D$ of a $(d+1)$-dimensional
Brownian motion from
an unbounded open domain
$D=\set{ (x, y) \in \R^{d+1} \dvtx y > f(x), x \in \R^d }$
starting at\vspace{0.5pt} $(x_0, f(x_0)+1)\in \R^{d+1}$ for some $x_0 \in \R^d$,
where the function $f(x)$ on $\R^d$ is convex and $f(x) \to \infty$ as
the Euclidean norm $|x| \to \infty$. Very general estimates for the
asymptotics of $\log \pr{\tau_D >t}$
are given by using Gaussian techniques.
In particular, for $f(x)=\exp\{ |x|^p\}$, $p >0$,
\[
\lim_{t\to\infty} t^{-1} (\log t)^{2/p} \log \pr{\tau_D>t}=-j_\nu^2/2,
\]
where $\nu=(d-2)/2$ and
$j_\nu$ is the smallest positive zero of the Bessel function $J_{\nu}$.