  Suppose $X$ is a Markov process on the real line (or some interval). Do the
distributions of its first passage times downwards (fptd) determine its law? In
this paper we treat some special cases of this question. We prove that if the
fptd process has the law of a subordinator, then necessarily $X$ is a L\'evy
process with no negative jumps; specifying the law of the subordinator
determines the law of $X$ uniquely. We further show that, likewise, the classes
of continuous-state branching processes and of self-similar processes without
negative jumps are also respectively characterised by a certain structure of
their fptd distributions; and each member of these classes separately is
determined uniquely by the precise family of its fptd laws. The road to these
results is paved by (i) the identification of Markov processes without negative
jumps in terms of the nature of their fptd laws, and (ii) some general results
concerning the identification of the fptd distributions for such processes.
