  Suppose that $(X_t)_{t \ge 0}$ is a one-dimensional Brownian motion with
negative drift $-\mu$. It is possible to make sense of conditioning this
process to be in the state $0$ at an independent exponential random time and if
we kill the conditioned process at the exponential time the resulting process
is Markov. If we let the rate parameter of the random time go to $0$, then the
limit of the killed Markov process evolves like $X$ conditioned to hit $0$,
after which time it behaves as $X$ killed at the last time $X$ visits $0$.
Equivalently, the limit process has the dynamics of the killed "bang--bang"
Brownian motion that evolves like Brownian motion with positive drift $+\mu$
when it is negative, like Brownian motion with negative drift $-\mu$ when it is
positive, and is killed according to the local time spent at $0$.
  An extension of this result holds in great generality for Borel right
processes conditioned to be in some state $a$ at an exponential random time, at
which time they are killed. Our proofs involve understanding the Campbell
measures associated with local times, the use of excursion theory, and the
development of a suitable analogue of the "bang--bang" construction for general
Markov processes.
  As examples, we consider the special case when the transient Borel right
process is a one-dimensional diffusion. Characterizing the limiting conditioned
and killed process via its infinitesimal generator leads to an investigation of
the $h$-transforms of transient one-dimensional diffusion processes that goes
beyond what is known and is of independent interest.
