Probability Surveys

On temporally completely monotone functions for Markov processes

Francis Hirsch and Marc Yor

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Any negative moment of an increasing Lamperti process(Yt,t0) is a completely monotone function of t. This property enticed us to study systematically, for a given Markov process (Yt,t0), 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.

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Probab. Surveys, Volume 9 (2012), 253-286.

First available in Project Euclid: 10 May 2012

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Primary: 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 60J25: Continuous-time Markov processes on general state spaces
Secondary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60G18: Self-similar processes

Temporally completely monotone function completely excessive function completely superharmonic function Lamperti’s correspondence Lamperti process Markov process


Hirsch, Francis; Yor, Marc. On temporally completely monotone functions for Markov processes. Probab. Surveys 9 (2012), 253--286. doi:10.1214/11-PS179.

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