  We address diffusion processes in a bounded domain, while focusing on
somewhat unexplored affinities between the presence of absorbing and/or
inaccessible boundaries. For the Brownian motion (L\'{e}vy-stable cases are
briefly mentioned) model-independent features are established, of the dynamical
law that underlies the short time behavior of these random paths, whose overall
life-time is predefined to be long. As a by-product, the limiting regime of a
permanent trapping in a domain is obtained. We demonstrate that the adopted
conditioning method, involving the so-called Bernstein transition function,
works properly also in an unbounded domain, for stochastic processes with
killing (Feynman-Kac kernels play the role of transition densities), provided
the spectrum of the related semigroup operator is discrete. The method is shown
to be useful in the case, when the spectrum of the generator goes down to zero
and no isolated minimal (ground state) eigenvalue is in existence, like e.g. in
the problem of the long-term survival on a half-line with a sink at origin.
