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On temporally completely monotone functions for Markov processes. (English) Zbl 1245.60071
Summary: Any negative moment of an increasing Lamperti process $$(Y_{t},t\geq 0)$$ is a completely monotone function of $$t$$. This property enticed us to study systematically for a given Markov process ($$Y_{t},t\geq 0$$) the functions $$f$$ such that the expectation of $$f(Y_{t})$$ 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 H. Ben Saad and K. Janßen [“Bernstein’s theorem for completely excessive measures”, Nagoya Math. J. 119, 133–141 (1990), \urleuclid:nmj/1118782040] 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.

##### MSC:
 60J45 Probabilistic potential theory 60J25 Continuous-time Markov processes on general state spaces 60J35 Transition functions, generators and resolvents 60G18 Self-similar stochastic processes
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##### References:
 [1] Ben Saad, H. and Janßen, K. (1990). Bernstein’s theorem for completely excessive measures. Nagoya Math. J. 119 133-141. [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. · Zbl 1024.60030 · doi:10.1214/ECP.v6-1039 · emis:journals/EJP-ECP/EcpVol6/paper10.abs.html · eudml:123012 [3] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probability Surveys 2 191-212. · Zbl 1189.60096 · doi:10.1214/154957805100000122 · eudml:228237 [4] Beznea, L. (1988). Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels. Nagoya Math. J. 112 125-142. · Zbl 0632.31008 [5] Carmona, P., Petit, F. and Yor, M. (1994). On exponential functionals of certain Lévy processes. Stochastics and Stochastic Rep. 47 71-101. · Zbl 0830.60072 [6] Chaumont, L. and Yor, M. (2003). Exercises in Probability. A guided tour from measure theory to random processes, via conditioning . Cambridge University Press. · Zbl 1065.60001 [7] Choquet, G. and Deny, J. (1960). Sur l’équation de convolution \mu = \mu * \sigma . C. R. Acad. Sc. Paris 250 799-801. · Zbl 0093.12802 [8] Deny, J. (1960). Sur l’équation de convolution \mu = \mu * \sigma . In Séminaire de Théorie du Potentiel (Brelot, Choquet, Deny), 4e année: 1959/60 , exposé n o 5, 11p. · Zbl 0151.16303 [9] Dynkin, E.B. (1980). Minimal excessive measures and functions. Trans. Amer. Math. Soc. 258-1 217-244. · Zbl 0422.60057 · doi:10.2307/1998292 [10] Getoor, R.K. (1975). On the construction of kernels. In Séminaire de Probabilités IX , Lect. Notes Math. 465 , Springer, 443-463. · Zbl 0321.60056 · numdam:SPS_1975__9__443_0 · eudml:113047 [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 . · Zbl 1287.60048 [12] Itô, K. and Mc Kean, H.P. (1974). Diffusion processes and their simple paths . Springer. [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. · Zbl 0427.31009 [14] Kunita, H. (1969). Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 1-26. · Zbl 0186.51203 [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. · Zbl 1246.60073 · doi:10.1214/EJP.v17-1755 [16] Lamperti, J. (1972). Semi-stable Markov processes. Zeit. für Wahr. 22-3 205-225. · Zbl 0274.60052 · doi:10.1007/BF00536091 [17] Lebedev, N.N. (1972). Special functions and their applications . Dover Publications. · Zbl 0271.33001 [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. · Zbl 0332.60032 · numdam:SPS_1976__10__125_0 · numdam:SPS_1976__10__142_0 · numdam:SPS_1976__10__164_0 · numdam:SPS_1976__10__175_0 [19] Patie, P. (2011). A refined factorization of the exponential law. Bernoulli 17-2 814-826. · Zbl 1253.60020 · doi:10.3150/10-BEJ292 [20] Pardo,J.C., Patie, P. and Savov, M. (2011). A Wiener-Hopf type factorization of the exponential functional of Lévy processes. arXiv: · Zbl 1272.60027 · arxiv.org · arxiv:1105.0062v2, [21] Revuz, D. and Yor, M. (1999). Continuous martingales and Brownian motion (third edition). Springer. · Zbl 0917.60006 [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. · Zbl 1082.60073 · doi:10.1016/j.anihpb.2004.01.006 · numdam:AIHPB_2005__41_3_335_0 · eudml:77848 [23] Tortrat, A. (1988). Le support des lois indéfiniment divisibles dans un groupe abélien localement compact. Math. Zeitschrift 197 231-250. · Zbl 0618.60013 · doi:10.1007/BF01215192 · eudml:183729 [24] Yan, J.-A. (1988). A formula for densities of transition functions. In Séminaire de Probabilités XXII , Lect. Notes Math. 1321 , Springer, 92-100. · Zbl 0662.60081 · numdam:SPS_1988__22__92_0 · eudml:113661 [25] Zolotarev, V.M. (1986). One-dimensional stable distributions . Amer. Math. Soc. · Zbl 0589.60015
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