Home | Current | Past volumes | About | Login | Notify | Contact | Search
 Probability Surveys > Vol. 9 (2012) open journal systems 

On temporally completely monotone functions for Markov processes

Francis Hirsch, University of Evry
Marc Yor, University Paris VI

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.

AMS 2000 subject classifications: Primary 60J45, 60J25; secondary 60J35, 60G18.

Keywords: Temporally completely monotone function, completely excessive function, completely superharmonic function, Lamperti's correspondence, Lamperti process, Markov process.

Creative Common LOGO

Full Text: PDF

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


[1]    Ben Saad, H. and Janssen, K. (1990). Bernstein’s theorem for completely excessive measures. Nagoya Math. J. 119 133–141. MR1071904

[2]    Bertoin, J. and Yor, M. (2001). On subordinators, self-similar Markov processes and some factorizations of the exponential variable. Elect. Comm. in Probab. 6 95–106. MR1871698

[3]    Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probability Surveys 2 191–212. MR2178044

[4]    Beznea, L. (1988). Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels. Nagoya Math. J. 112 125–142. MR0974268

[5]    Carmona, P., Petit, F. and Yor, M. (1994). On exponential functionals of certain Lévy processes. Stochastics and Stochastic Rep. 47 71–101. MR1787143

[6]    Chaumont, L. and Yor, M. (2003). Exercises in Probability. A guided tour from measure theory to random processes, via conditioning. Cambridge University Press. MR2016344

[7]    Choquet, G. and Deny, J. (1960). Sur l’équation de convolution μ = μ σ. C. R. Acad. Sc. Paris 250 799–801. MR0119041

[8]    Deny, J. (1960). Sur l’équation de convolution μ = μσ. In Séminaire de Théorie du Potentiel (Brelot, Choquet, Deny), 4e année: 1959/60, exposé no 5, 11p.

[9]    Dynkin, E.B. (1980). Minimal excessive measures and functions. Trans. Amer. Math. Soc. 258-1 217–244. MR0554330

[10]    Getoor, R.K. (1975). On the construction of kernels. In Séminaire de Probabilités IX, Lect. Notes Math. 465, Springer, 443–463. MR0436342

[11]    Hirsch, F. and Yor, M. (2011). On the remarkable Lamperti representation of the inverse local time of a radial Ornstein-Uhlenbeck process. Prépublication 324, 10/2011, Université d’Evry.

[12]    Itô, K. and Mc Kean, H.P. (1974). Diffusion processes and their simple paths. Springer. MR0345224

[13]    Itô, M. and Suzuki, N. (1981). Completely superharmonic measures for the infinitesimal generator A of a diffusion semi-group and positive eigen elements of A. Nagoya Math. J. 83 53–106. MR0632647

[14]    Kunita, H. (1969). Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 1–26. MR0250387

[15]    Kuznetsov, A., Pardo, J.C. and Savov, M. (2012). Distributional properties of exponential functionals of Lévy processes. Electron. J. Probab. 17-8 1–35.

[16]    Lamperti, J. (1972). Semi-stable Markov processes. Zeit. für Wahr. 22-3 205–225. MR0307358

[17]    Lebedev, N.N. (1972). Special functions and their applications. Dover Publications. MR0350075

[18]    Meyer, P.-A. (1976). Démonstration probabiliste de certaines inégalités de Littlewood-Paley. Exposé II: l’opérateur carré du champ. In Séminaire de Probabilités X, Lect. Notes Math. 511, Springer, 142–161. MR0501380

[19]    Patie, P. (2011). A refined factorization of the exponential law. Bernoulli 17-2 814–826. MR2787616

[20]    Pardo,J.C., Patie, P. and Savov, M. (2011). A Wiener-Hopf type factorization of the exponential functional of Lévy processes. arxiv:1105.0062v2, 2011.

[21]    Revuz, D. and Yor, M. (1999). Continuous martingales and Brownian motion (third edition). Springer. MR1725357

[22]    Salminen, P. and Yor, M. (2005). Properties of perpetual integral functionals of Brownian motion with drift. Ann. Inst. H. Poincaré (B) Probability and Statistics 41-3 335–347. MR2139023

[23]    Tortrat, A. (1988). Le support des lois indéfiniment divisibles dans un groupe abélien localement compact. Math. Zeitschrift 197 231–250. MR0923491

[24]    Yan, J.-A. (1988). A formula for densities of transition functions. In Séminaire de Probabilités XXII, Lect. Notes Math. 1321, Springer, 92–100. MR0960514

[25]    Zolotarev, V.M. (1986). One-dimensional stable distributions. Amer. Math. Soc. MR0854867

Home | Current | Past volumes | About | Login | Notify | Contact | Search

Probability Surveys. ISSN: 1549-5787