On temporally completely monotone functions for Markov processes

Any negative moment of an increasing Lamperti process(Yt,t≥0) is a completely monotone function of t. This property enticed us to study systematically, for a given Markov process (Yt,t≥0), the functions f such that the expectation of f(Yt) is a completely monotone function of t. We call these functions temporally completely monotone (for Y). Our description of these functions is deduced from the analysis made by Ben Saad and Janßen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a Lévy process, a Bessel process, or an increasing Lamperti process.