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

Some characterizations for Markov processes at first passage