Bounds for Kac's master equation

Communications in Mathematical Physics

Volumn 209, Issue 3, 2000, Pages 729-755

  Diaconis, Persi ^a   Saloff Coste, Laurent ^b,c  

^a STANFORD UNIVERSITY   (United States)

^b CNRS   (France)

^c Department of Population Medicine and Diagnostic Sciences   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034348847     PISSN: 00103616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002200050036     Document Type: Article

References (44)

1. th S.T.O.C., Assoc. Comp. Mach. New York, pp. 176-188
2. 2. Bird, G. A. (1976): Molecular Gas Dynamics Oxford: Clarendon Press,
3. 3. Carlen, E., Gabetta, E. and Toscani, G. (1997): Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Preprint, Dept. of Mathematics, Georgia Tech.
4. 4. Cercignani, C., Illner, R. and Pulverinti, M. (1994): The mathematical theory of dilute gases. New York: Springer
5. 5. Desvillettes, L. (1995): About the regularizing properties of the non cutoff Kac equation. Commun. Math. Phys. 168, 416-440
6. 6. Diaconis, P. and Mallows, C. (1997): Identities for normal variables arising from the classical groups. Technical Report, Dept. of Mathematics, Cornell University
7. 7. Diaconis, P. and Saloff-Coste, L. (1993 a): Comparison techniques for random walk on finite groups. Ann. Probab. 21, 2131-2156
8. 8. Diaconis, P. and Saloff-Coste, L. (1993 b): Comparison theorems for reversible Markov chains. Ann. Appl. Math. 3, 696-730
9. 9. Diaconis, P. and Shahshahani, M. (1996): Products of random matrices as they arise in the study of random walks on groups. Contemp.Math. 50, 183-195
10. 10. Diaconis, P. and Shahshahani, M. (1987): The subgroup algorithm for generating uniform random variables. Prob. Eng. Info. Sci. 1, 15-32
11. 11. Dudley, R. (1966): Convergence of Baire measures. Studia Math. 27, 251-268
12. 12. Dudley, R. (1968): Distances of probability measures and random variables. Ann. Math. Statist. 39, 1563-1572
13. 13. Dudley, R. (1976): Probabilities and Metrics. Lecture Notes Series 45, Matematisk Institut Aarhus
14. 14. Dudley, R. (1989): Real Analysis and Probability Pacific Grove, CA: Wadsworth & Brooks/Cole
15. 15. Gallot, S., Hulin, D. and Lafontaine, J. (1990): Riemannian Geometry. 2nd ed., Berlin: Springer-Verlag
16. 16. Gottlieb, A. L. (1998): Markov Transitions and Propagation of Chaos. Ph.D thesis, Department of Mathematics. University of California, Berkeley
17. 17. Grünbaum, F. A. (1971): Propagation of chaos for the Boltzmann equation. Arch. Rat. Mech. and Anal. 42, 323-345
18. 18. Grünbaum, A. (1972): Linearization for the Boltzmann equation. Trans. Amer. Math. Soc. 165, 425-499
19. 19. Hastings, W. (1970): Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97-109
20. 20. Hurwitz, A. (1897): Über die Erzeugung der Invarianten durch Integration. Nach. Gesell. Wissen, Göttingen Math-Phys Klasse, pp. 71-90; Reprinted in Hurwitz, A. (1963): Mathematische Werke, Vol. II, Basel: Birkhäuser, pp. 546-564
21. 20. Hurwitz, A. (1897): Über die Erzeugung der Invarianten durch Integration. Nach. Gesell. Wissen, Göttingen Math-Phys Klasse, pp. 71-90; Reprinted in Hurwitz, A. (1963): Mathematische Werke, Vol. II, Basel: Birkhäuser, pp. 546-564
22. 21. Janvresse, E. (1999): Bounds on random rotations on the sphere. Preprint
23. 22. Kac, M. (1956): Foundations of Kinetic Theory. Proc. 3rd Berkeley Sympos., J. Neymann, ed. Vol. 3, pp. 171-197
24. 23. Kac, M. (1959): Probability and Related Topics in Physical Science. New York: Wiley Interscience
25. 24. Kloss, B. (1959): Limiting distributions on bicompact topological groups. Th. Prob. Appl. 4, 237-270
26. 25. Kuipers, L. and Niederreiter, H. (1974): Uniform Distribution of Sequences. New York: Wiley
27. 26. Lubotzky, A., Phillips, R. and Sarnak, P. (1986): Hecke operators and distributing points on the sphere I. Comm. Pure Appl. Math. 39, Supplement 1, S149-S186
28. 27. Maslin, D. (1999): The eigenvalues of Kac's master equation. Preprint, Department of Mathematics, Dartmouth
29. 28. McKean, H. (1966): Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rat. Mech. Anal. 2, 343-367
30. 29. Méléard, S. (1996): Asymptotic behavior of some interacting particle systems, Mc Kean-Vlasov and Boltzmann models. Lecture Notes in Math. 1627, New York: Springer
31. 30. Nambu, K. (1983): Interrelations between various direct simulation methods for solving the Boltzmann equation. J. Phys. soc. Japan 52, 3382-3388
32. 31. Perthame, B. (1994): Introduction to the theory of random particles methods for Boltzmann equation. In: Progress on Kinetic Theory, Singapore: World Scientific
33. 2-lower bounds for a special class of random walks. Probab. Th. Related Fields 101, 277-289
34. 33. Porod, U. (1996 a): The cut-off phenomenon for random reflections. Ann. Prob. 24, 74-99
35. 34. Porod, U. (1996 b): The cut-off phenomenon for random reflections II: Complex and quaternionic cases. Probab. Th. Related Fields 104, 181-209
36. 35. Riesz, F., Nagy, B. (1960): Functional Analysis. Unger, New York
37. 36. Rosenthal, J. (1994): Random rotations: Characters and random walks on SO(n). Ann. Probab. 22, 398-423
38. 37. Rothaus, O. (1981): Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities. J. Func. Anal. 42, 102-109
39. 38. Saloff-Coste, L. (1997): Lectures on finite Markov chains. Ecole d'ete de St. Flour, LNM 1665, Berlin-Heidelberg-New York: Springer
40. 39. Shor, P. (1998): Quantum computation. Proc. I.C.M., Berlin 1998, I, 467-486
41. 40. Smith, A. and Roberts, G. (1993): Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods (with discussion). J. Roy. Stat. Soc. B 55, 3-24
42. 41. Su, F. (1995): Ph. D. Dissertation Dept. Mathematics, Harvard University
43. 42. Sznitman, A.-S. (1991): Topics in Propagation of Chaos. Springer Lecture Notes in Math, 1464, New York: Springer
44. 43. Uchiyama, K. (1988): Derivation of the Boltzmann equation from particle dynamics. Hiroshima Math. J. 18, 245-297