On the coupling property of Lévy processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1147-1159.

Nous donnons les conditions nécessaires et suffisantes pour le succès du couplage entre des processus de Lévy (avec partie de sauts non-dégénérée). Notre méthode est basée sur les formules explicites pour le semigroupe de transition d'un processus de Poisson composé, et les résultats de Mineka et Lindvall-Rogers sur le couplage d'une marche aléatoire. En particulier, nous montrons qu'un processus de Lévy admet un couplage, s'il est un processus fortement fellerien ou si la mesure de Lévy (mesure de sauts) possède une composante absolument continue.

We give necessary and sufficient conditions guaranteeing that the coupling for Lévy processes (with non-degenerate jump part) is successful. Our method relies on explicit formulae for the transition semigroup of a compound Poisson process and earlier results by Mineka and Lindvall-Rogers on couplings of random walks. In particular, we obtain that a Lévy process admits a successful coupling, if it is a strong Feller process or if the Lévy (jump) measure has an absolutely continuous component.

DOI : 10.1214/10-AIHP400
Classification : 60G51, 60G50, 60J25, 60J75
Mots-clés : coupling property, Lévy processes, compound Poisson processes, random walks, Mineka coupling, strong Feller property 
@article{AIHPB_2011__47_4_1147_0,
     author = {Schilling, Ren\'e L. and Wang, Jian},
     title = {On the coupling property of {L\'evy} processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1147--1159},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {4},
     year = {2011},
     doi = {10.1214/10-AIHP400},
     zbl = {1268.60061},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP400/}
}
TY  - JOUR
AU  - Schilling, René L.
AU  - Wang, Jian
TI  - On the coupling property of Lévy processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
SP  - 1147
EP  - 1159
VL  - 47
IS  - 4
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/10-AIHP400/
DO  - 10.1214/10-AIHP400
LA  - en
ID  - AIHPB_2011__47_4_1147_0
ER  - 
%0 Journal Article
%A Schilling, René L.
%A Wang, Jian
%T On the coupling property of Lévy processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 1147-1159
%V 47
%N 4
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/10-AIHP400/
%R 10.1214/10-AIHP400
%G en
%F AIHPB_2011__47_4_1147_0
Schilling, René L.; Wang, Jian. On the coupling property of Lévy processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1147-1159. doi : 10.1214/10-AIHP400. http://www.numdam.org/articles/10.1214/10-AIHP400/

[1] B. Böttcher, R. L. Schilling and J. Wang. Constructions of coupling processes for Lévy processes. Stochastic Process. Appl. To appear (2011). Available at arXiv:1009.5511v1. | MR | Zbl

[2] M. F. Chen. Eigenvalues, Inequalities and Ergodic Theory. Springer, London, 2005. | MR | Zbl

[3] M. Cranston and A. Greven. Coupling and harmonic functions in the case of continuous time Markov processes. Stochastic Process. Appl. 60 (1995) 261-286. | MR | Zbl

[4] M. Cranston and F.-Y. Wang. A condition for the equivalence of coupling and shift-coupling. Ann. Probab. 28 (2000) 1666-1679. | MR | Zbl

[5] J. Hawkes. Potential theory of Lévy processes. Proc. London Math. Soc. 38 (1979) 335-352. | MR | Zbl

[6] N. Jacob. Pseudo Differential Operators and Markov Processes. Volume 1: Fourier Analysis and Semigroups. Imperial College Press, London, 2001. | MR | Zbl

[7] T. Lindvall. Lectures on the Coupling Method. Wiley, New York, 1992. | MR | Zbl

[8] T. Lindvall and L. C. G. Rogers. On the coupling of random walks and renewal processes. J. Appl. Probab. 33 (1996) 122-126. | MR | Zbl

[9] J. Mineka. A criterion for tail events for sums of independent random variables. Z. Wahrsch. Verw. Gebiete 25 (1973) 163-170. | MR | Zbl

[10] D. Revuz. Markov Chains, 2nd edition. North-Holland Mathematical Library 11. North-Holland, Netherlands, 1984. | MR | Zbl

[11] K. Sato. Lévy Processes and Infinitely Divisible Distributions. Studies Adv. Math. 68. Cambridge Univ. Press, Cambridge, 1999. | MR | Zbl

[12] H. Thorisson. Shift-coupling in continuous time. Probab. Theory Related Fields 99 (1994) 477-483. | MR | Zbl

[13] H. Thorisson. Coupling, Stationarity and Regeneration. Springer, New York, 2000. | MR | Zbl

[14] F.-Y. Wang. Coupling for Ornstein-Uhlenbeck jump processes. Bernoulli (2010). To appear. Available at arXiv:1002.2890v5. | MR

[15] F.-Y. Wang. Functional Inequalities, Markov Processes and Spectral Theory. Science Press, Beijing, 2005. | MR

Cité par Sources :