Lipschitzian norm estimate of one-dimensional Poisson equations and applications
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 450-465.

By direct calculus we identify explicitly the lipschitzian norm of the solution of the Poisson equation in terms of various norms of g, where is a Sturm-Liouville operator or generator of a non-singular diffusion in an interval. This allows us to obtain the best constant in the L1-Poincaré inequality (a little stronger than the Cheeger isoperimetric inequality) and some sharp transportation-information inequalities and concentration inequalities for empirical means. We conclude with several illustrative examples.

Par un calcul direct, on identifie explicitement la norme Lipschitzienne de la solution de l'équation de Poisson en terme de différentes normes de g, où est l'opérateur de Sturm-Liouville ou le générateur d'une diffusion non singulière sur un intervalle. Ainsi, nous pouvons obtenir, d'une part la meilleure constante dans l'inégalité de Poincaré L1 (une inégalité un peu plus forte que l'inégalité isopérimétrique de Cheeger) et d'autre part certaines inégalités de transport-information et de concentration fines pour la moyenne empirique. On conclut avec des exemples illustratifs.

DOI: 10.1214/10-AIHP360
Classification: 47B38,  60E15,  60J60,  34L15,  35P15
Keywords: Poisson equations, transportation-information inequalities, concentration and isoperimetric inequalities
@article{AIHPB_2011__47_2_450_0,
     author = {Djellout, Hacene and Wu, Liming},
     title = {Lipschitzian norm estimate of one-dimensional {Poisson} equations and applications},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {450--465},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {2},
     year = {2011},
     doi = {10.1214/10-AIHP360},
     zbl = {1233.47029},
     mrnumber = {2814418},
     language = {en},
     url = {http://www.numdam.org/articles/10.1214/10-AIHP360/}
}
TY  - JOUR
AU  - Djellout, Hacene
AU  - Wu, Liming
TI  - Lipschitzian norm estimate of one-dimensional Poisson equations and applications
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2011
DA  - 2011///
SP  - 450
EP  - 465
VL  - 47
IS  - 2
PB  - Gauthier-Villars
UR  - http://www.numdam.org/articles/10.1214/10-AIHP360/
UR  - https://zbmath.org/?q=an%3A1233.47029
UR  - https://www.ams.org/mathscinet-getitem?mr=2814418
UR  - https://doi.org/10.1214/10-AIHP360
DO  - 10.1214/10-AIHP360
LA  - en
ID  - AIHPB_2011__47_2_450_0
ER  - 
%0 Journal Article
%A Djellout, Hacene
%A Wu, Liming
%T Lipschitzian norm estimate of one-dimensional Poisson equations and applications
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2011
%P 450-465
%V 47
%N 2
%I Gauthier-Villars
%U https://doi.org/10.1214/10-AIHP360
%R 10.1214/10-AIHP360
%G en
%F AIHPB_2011__47_2_450_0
Djellout, Hacene; Wu, Liming. Lipschitzian norm estimate of one-dimensional Poisson equations and applications. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 2, pp. 450-465. doi : 10.1214/10-AIHP360. http://www.numdam.org/articles/10.1214/10-AIHP360/

[1] F. Barthe and A. V. Kolesnikov. Mass transport and variants of the logarithmic Sobolev inequality. J. Geom. Anal. 18 (2008) 921-979. | MR | Zbl

[2] F. Barthe and C. Roberto. Sobolev inequalities for probability measures on the real line. Studia Math. 159 (2003) 481-497. | 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] S. G. Bobkov and C. Houdre. Some connections between isoperimetric and Sobolev-type inequalities. Mem. Amer. Math. Soc. 129 (1997) No. 616. | MR | Zbl

[5] S. G. Bobkov and C. Houdre. Isoperimetric constants for product probability measures. Ann. Probab. 25 (1997) 184-205. | MR | Zbl

[6] S. G. Bobkov and M. Ledoux. Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37 (2009) 403-427. | MR | Zbl

[7] P. Buser. A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. 15 (1982) 213-230. | Numdam | MR | Zbl

[8] M. F. Chen. Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. China Ser. A 42 (1999) 805-815. | MR | Zbl

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

[10] N. Demni and M. Zani. Large deviations for statistics of the Jacobi process. Stochastic Process. Appl. 119 (2009) 518-533. | MR | Zbl

[11] H. Djellout. Lp-uniqueness for one-dimensional diffusions. In Mémoire de D.E.A. Université Blaise Pascal, Clermont-Ferrand, 1997.

[12] A. Eberle. Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators. Lecture Notes in Mathematics 1718. Springer, Berlin, 1999. | MR | Zbl

[13] N. Gozlan, Poincaré inequalities and dimension free concentration of measure. Ann. Inst. H. Poincaré Probab. Statist. To appear. | EuDML | Numdam | MR | Zbl

[14] N. Gozlan and C. Léonard. A large deviation approach to some transportation cost inequalities. Probab. Theory Related Fields 139 (2007) 235-283. | MR | Zbl

[15] A. Guillin, C. Léonard, L. Wu and N. Yao. Transport-information inequalities for Markov processes (I). Probab. Theory Related Fields 144 (2009) 669-695. | MR | Zbl

[16] A. Guillin, C. Léonard, F. Y. Wang and L. Wu. Transportation-information inequalities for Markov processes (II): Relations with other functional inequalities. Preprint. Available at http://arxiv.org/abs/0902.2101 or http://hal.archives-ouvertes.fr/hal-00360854/fr/. | Zbl

[17] N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd edition. North-Holland Mathematical Library 24. North-Holland, Amsterdam, 1989. | MR | Zbl

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

[19] O. Ludger. Estimation for continuous branching processes. Scand. J. Statist. 25 (1998) 111-126. | MR | Zbl

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

[21] M. Ledoux. Spectral gap, logarithmic Sobolev constant, and geometric bounds. Surv. Differ. Geom. IX (2004) 219-240. | MR | Zbl

[22] E. Milman. On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (2009) 1-43. | MR | Zbl

[23] 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

[24] F. Y. Wang. Functional Inequalities, Markov Semigroup and Spectral Theory. Chinese Sciences Press, Beijing, 2005.

[25] L. M. Wu. Moderate deviations of dependent random variables related to CLT. Ann. Probab. 23 (1995) 420-445. | MR | Zbl

[26] L. Wu, Gradient estimates of Poisson equations on Riemannian manifolds and applications. J. Funct. Anal. 29 (2009) 1008-1022. | MR | Zbl

[27] L. Wu and Y. P. Zhang. A new topological approach to the L∞-uniqueness of operators and the L1-uniqueness of Fokker-Planck equations. J. Funct. Anal. 241 (2006) 557-610. | MR | Zbl

Cited by Sources: