Probability Theory and Related Fields

, Volume 140, Issue 1–2, pp 19–40 | Cite as

Probabilistic approach for granular media equations in the non-uniformly convex case

Article

Abstract

We use here a particle system to prove both a convergence result (with convergence rate) and a deviation inequality for solutions of granular media equation when the confinement potential and the interaction potential are no more uniformly convex. The proof of convergence is simpler than the one in Carrillo–McCann–Villani (Rev. Mat. Iberoamericana 19:971–1018, 2003; Arch. Rat. Mech. Anal. 179:217–263, 2006). All the results complete former results of Malrieu (Ann. Appl. Probab. 13:540–560, 2003) in the uniformly convex case. The main tool is an uniform propagation of chaos property and a direct control in Wasserstein distance of solutions starting with different initial measures. The deviation inequality is obtained via a T 1 transportation cost inequality replacing the logarithmic Sobolev inequality which is no more clearly dimension free.

Keywords

Granular media equation Transportation cost inequality Logarithmic Sobolev Inequalities Concentration inequalities 

Mathematics Subject Classification (2000)

65C35 35K55 65C05 82C22 26D10 60E15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benachour S., Roynette B., Talay D. and Vallois P. (1998). Nonlinear self-stabilizing processes. I. Existence, invariant probability, propagation of chaos. Stochastic Process. Appl. 75(2): 173–201 MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benachour S., Roynette B. and Vallois P. (1998). Nonlinear self-stabilizing processes. II. Convergence to invariant probability. Stochastic Process. Appl. 75(2): 203–224 MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Benedetto D., Caglioti E. and Pulvirenti M. (1997). A kinetic equation for granular media equation. Rairo Modél. Math. Anal. Num. 31(5): 615–641 MATHMathSciNetGoogle Scholar
  4. 4.
    Benedetto D., Caglioti E., Carillo J.A. and Pulvirenti M. (1998). A non Maxwellian steady distribution for one-dimensional granular media. J. Stat. Phys. 91(5/6): 979–990 MATHCrossRefGoogle Scholar
  5. 5.
    Bolley F. and Villani C. (2005). Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Facult. Sci. Toulouse 6: 331–352 MathSciNetGoogle Scholar
  6. 6.
    Bolley, F., Guillin, A., Villani, C.: Quantitative concentration inequalities for empirical measures on noncompact spaces. Prob. Theor. Rel. Fields. (in press)Google Scholar
  7. 7.
    Carrillo J.A., McCann R.J. and Villani C. (2003). Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev. Mat. Iberoamericana 19(3): 971–1018 MATHMathSciNetGoogle Scholar
  8. 8.
    Carrillo J.A., McCann R.J. and Villani C. (2006). Contractions in the 2-Wasserstein length space and thermalization of granular media. Arch. Rat. Mech. Anal. 179(2): 217–263 MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cattiaux, P., Léonard, C.: Minimization of the Kullback information of diffusion processes. Ann. Inst. Henri Poincaré 30(1), 83–132 (1994), and correction in Ann. Inst. Henri Poincaré 31, 705–707 (1995)Google Scholar
  10. 10.
    Djellout H., Guillin A. and Wu L. (2004). Transportation cost-information inequalities and applications to random dynamical systems and diffusions. Ann. Probab. 32(3B): 2702–2732 MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Down D., Meyn S.P. and Tweedie R.L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4): 1671–1691 MATHMathSciNetGoogle Scholar
  12. 12.
    Gozlan, N.: Principe conditionnel de Gibbs pour des contraintes fines approchées et inégalités de transport. PhD Thesis, Université de Paris 10. Available online via http://tel.ccsd.cnrs.fr/documents/archives0/00/01/01/73/tel-00010173-00/tel-00010173-00.pdf (2005)Google Scholar
  13. 13.
    Lamba, H., Mattingly, J.C., Stuart, A.: An adaptive Euler–Maruyama scheme for SDEs: convergence and stability. Preprint. Available online via http://front.math.ucdavis.edu/math.NA/0601029 (2006)Google Scholar
  14. 14.
    Malrieu F. (2001). Logarithmic Sobolev inequalities for some nonlinear PDE’s. Stochastic Process. Appl. 95(1): 109–132 MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Malrieu F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13(2): 540–560 MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Méléard, S.: Asymptotic behaviour of some interacting particle systems: Mc Kean-Vlasov and Boltzmann models. In: Probabilistic Models for Nonlinear Partial Differential Equations. Talay and Tubaro, Lecture Notes in Mathematics, vol 1627, pp. 42–95 (1995)Google Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.École Polytechnique, CMAPPalaiseau CedexFrance
  2. 2.Université Paris X Nanterre, Equipe MODAL’X, UFR SEGMINanterre CedexFrance
  3. 3.CEREMADE, UMR CNRS 7534Paris Cedex 16France
  4. 4.IRMAR, Université Rennes 1Rennes CedexFrance

Personalised recommendations