On temporally completely monotone functions for Markov processes

Any negative moment of an increasing Lamperti process (Xt ; 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 Janen, 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 L evy process, a Bessel process, or an increasing Lamperti process.