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

Probability Theory and Related Fields

Volumn 140, Issue 1-2, 2008, Pages 19-40

  Cattiaux, P ^a,b   Guillin, A ^c   Malrieu, F ^d  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^c Centre National de la Recherche Scientifique   (France)

^d UNIVERSITÉ DE RENNES 1   (France)

Author keywords

Concentration inequalities; Granular media equation; Logarithmic Sobolev Inequalities; Transportation cost inequality

Indexed keywords

EID: 36148968957     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-007-0056-3     Document Type: Article

References (16)

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
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
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
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
5. Bolley F. and Villani C. (2005). Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. Ann. Facult. Sci. Toulouse 6: 331-352
6. Bolley, F., Guillin, A., Villani, C.: Quantitative concentration inequalities for empirical measures on noncompact spaces. Prob. Theor. Rel. Fields. (in press)
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
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
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)
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
11. Down D., Meyn S.P. and Tweedie R.L. (1995). Exponential and uniform ergodicity of Markov processes. Ann. Probab. 23(4): 1671-1691
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)
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)
14. Malrieu F. (2001). Logarithmic Sobolev inequalities for some nonlinear PDE's. Stochastic Process. Appl. 95(1): 109-132
15. Malrieu F. (2003). Convergence to equilibrium for granular media equations and their Euler schemes. Ann. Appl. Probab. 13(2): 540-560
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)