  In this paper we review some results on time-homogeneous birth-death
processes. Specifically, for truncated birth-death processes with two absorbing
or two reflecting endpoints, we recall the necessary and sufficient conditions
on the transition rates such that the transition probabilities satisfy a
spatial symmetry relation. The latter leads to simple expressions for
first-passage-time densities and avoiding transition probabilities. This
approach is thus thoroughly extended to the case of bilateral birth-death
processes, even in the presence of catastrophes, and to the case of a
two-dimensional birth-death process with constant rates.
