no. 2
Transportation inequalities for stochastic differential equations of pure jumps
Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 465-479.

Pour une équation différentielle stochastique de pur saut, bien que l'inégalité de Poincaré ne soit pas valide en général, nous pouvons quand même établir, sous la condition de dissipativité, des inégalités de transport W1H pour sa mesure invariante et pour sa loi (au niveau de processus) sur l'espace des trajectoires càdlàg, muni de la métrique L1 ou d'une métrique uniforme. Quelques applications aux inégalités de concentration sont présentées.

For stochastic differential equations of pure jumps, though the Poincaré inequality does not hold in general, we show that W1H transportation inequalities hold for its invariant probability measure and for its process-level law on right continuous paths space in the L1-metric or in uniform metrics, under the dissipative condition. Several applications to concentration inequalities are given.

DOI : 10.1214/09-AIHP320
Classification : 60E15, 60H10, 60H07
Mots clés : transportation inequalities, stochastic differential equations, Malliavin calculus 
@article{AIHPB_2010__46_2_465_0,
     author = {Wu, Liming},
     title = {Transportation inequalities for stochastic differential equations of pure jumps},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {465--479},
     publisher = {Gauthier-Villars},
     volume = {46},
     number = {2},
     year = {2010},
     doi = {10.1214/09-AIHP320},
     mrnumber = {2667706},
     zbl = {1209.60015},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/09-AIHP320/}
}
TY  - JOUR
AU  - Wu, Liming
TI  - Transportation inequalities for stochastic differential equations of pure jumps
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2010
SP  - 465
EP  - 479
VL  - 46
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/09-AIHP320/
DO  - 10.1214/09-AIHP320
LA  - en
ID  - AIHPB_2010__46_2_465_0
ER  - 
%0 Journal Article
%A Wu, Liming
%T Transportation inequalities for stochastic differential equations of pure jumps
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2010
%P 465-479
%V 46
%N 2
%I Gauthier-Villars
%U http://www.numdam.org/articles/10.1214/09-AIHP320/
%R 10.1214/09-AIHP320
%G en
%F AIHPB_2010__46_2_465_0
Wu, Liming. Transportation inequalities for stochastic differential equations of pure jumps. Annales de l'I.H.P. Probabilités et statistiques, Tome 46 (2010) no. 2, pp. 465-479. doi : 10.1214/09-AIHP320. http://www.numdam.org/articles/10.1214/09-AIHP320/

[1] D. Bakry and M. Emery. Diffusions hypercontractives. In Séminaire de Probabilités 177-206. Lecture Notes in Math. 1123. Springer Berlin, 1985. | EuDML | Numdam | MR | Zbl

[2] K. Bichteler, J. B. Gravereaux and J. Jacod. Malliavin Calculus for Processes With Jumps. Stochastics Monographs 2. Gordon and Breach, New York, 1987. | MR | Zbl

[3] S. G. Bobkov and F. Götze. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1999) 1-28. | MR | Zbl

[4] F. Bolley, A. Guillin and C. Villani. Quantitative concentration inequalities for empirical measures on non compact spaces. Probab. Theory Related Fields 137 (2007) 541-593. | MR | Zbl

[5] H. Djellout, A. Guillin and L. Wu. Transportation cost-information inequalities for random dynamical systems and diffusions. Ann. Probab. 32 (2004) 2702-2732. | MR | Zbl

[6] I. Gentil and C. Imbert. The Lévy-Fokker-Planck equation: Φ-entropies and convergence to equilibrium. Preprint, 2008. | MR | Zbl

[7] N. Gozlan and C. Léonard. A large deviation approach to some transportation cost inequalities. Probab. Theory Related Fields. To appear, 2009. | MR | Zbl

[8] A. Guillin, C. Léonard, L. Wu and N. Yao. Transport-information inequalities for Markov processes (I). Probab. Theory Related Fields. Preprint, 2007. To appear. | MR | Zbl

[9] J. Jacod. The Euler scheme for Lévy driven stochastic differential equations: Limit theorems. Ann. Probab. 32 (2004) 1830-1872. | MR | Zbl

[10] T. Klein, Y. Ma and N. Privault. Convex concentration inequalities and forward-backward stochastic calculus. Electron. J. Probab. 11 (2006) 486-512. | MR | Zbl

[11] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. Amer. Math. Soc., Providence, RI, 2001. | MR | Zbl

[12] F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173 (2000) 361-400. | MR | Zbl

[13] J. Picard. Formules de dualité sur l'espace de Poisson. Ann. Inst. H. Poincaré Probab. Statist. 32 (1996) 509-548. | Numdam | MR | Zbl

[14] C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI, 2003. | MR | Zbl

[15] M. Röckner and F. Y. Wang. Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal. 203 (2003) 237-261. | MR | Zbl

[16] L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications. Probab. Theory Related Fields 118 (2000) 427-438. | MR | Zbl

[17] L. Wu. Uniformly integrable operators and large deviations for Markov processes. J. Funct. Anal. 172 (2000) 301-376. | MR | Zbl

[18] L. Wu. Essential spectral radius for Markov semigroups (I): Discrete time case. Probab. Theory Related Fields 128 (2004) 255-321. | MR | Zbl

Cité par Sources :