  We investigate the spectral properties of the Steklov problem for the
modified Helmholtz equation $(p-\Delta) u = 0$ in the exterior of a compact
set, for which the positive parameter $p$ ensures exponential decay of the
Steklov eigenfunctions at infinity. We obtain the small-$p$ asymptotic behavior
of the eigenvalues and eigenfunctions and discuss their features for different
space dimensions. These results find immediate applications to the theory of
stochastic processes and unveil the long-time asymptotic behavior of
probability densities of various first-passage times in exterior domains.
Theoretical results are validated by solving the exterior Steklov problem by a
finite-element method with a transparent boundary condition.
