Survey on Probabilistic Methods for the Study of Kac-like Equations

Federico Bassetti; Ester Gabetta

Bollettino dell'Unione Matematica Italiana (2011)

  • Volume: 4, Issue: 2, page 187-212
  • ISSN: 0392-4041

Abstract

top
This mainly explanatory paper shows how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the relaxation to equilibrium of the solutions of one-dimensional Boltzmann type equations. In particular, conditions under which the solutions of these equations converge to suitable scale mixture of stable distributions are reviewed. In addition, some recent results about the rate of convergence to steady states, with respect to various metrics, are summarized. Finally, by resorting to the above mentioned probabilistic methods, some new results related to a linear kinetic model are proven.

How to cite

top

Bassetti, Federico, and Gabetta, Ester. "Survey on Probabilistic Methods for the Study of Kac-like Equations." Bollettino dell'Unione Matematica Italiana 4.2 (2011): 187-212. <http://eudml.org/doc/290744>.

@article{Bassetti2011,
abstract = {This mainly explanatory paper shows how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the relaxation to equilibrium of the solutions of one-dimensional Boltzmann type equations. In particular, conditions under which the solutions of these equations converge to suitable scale mixture of stable distributions are reviewed. In addition, some recent results about the rate of convergence to steady states, with respect to various metrics, are summarized. Finally, by resorting to the above mentioned probabilistic methods, some new results related to a linear kinetic model are proven.},
author = {Bassetti, Federico, Gabetta, Ester},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {6},
number = {2},
pages = {187-212},
publisher = {Unione Matematica Italiana},
title = {Survey on Probabilistic Methods for the Study of Kac-like Equations},
url = {http://eudml.org/doc/290744},
volume = {4},
year = {2011},
}

TY - JOUR
AU - Bassetti, Federico
AU - Gabetta, Ester
TI - Survey on Probabilistic Methods for the Study of Kac-like Equations
JO - Bollettino dell'Unione Matematica Italiana
DA - 2011/6//
PB - Unione Matematica Italiana
VL - 4
IS - 2
SP - 187
EP - 212
AB - This mainly explanatory paper shows how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the relaxation to equilibrium of the solutions of one-dimensional Boltzmann type equations. In particular, conditions under which the solutions of these equations converge to suitable scale mixture of stable distributions are reviewed. In addition, some recent results about the rate of convergence to steady states, with respect to various metrics, are summarized. Finally, by resorting to the above mentioned probabilistic methods, some new results related to a linear kinetic model are proven.
LA - eng
UR - http://eudml.org/doc/290744
ER -

References

top
  1. ALSMEYER, G. - IKSANOV, A. - RÖSLER, U., On distributional properties of perpetuities. J. Theoret. Probab.22 (2009), 666-682. Zbl1173.60309MR2530108DOI10.1007/s10959-008-0156-8
  2. AMBROSIO, L. - GIGLI, N. - SAVARÉ, G. , Gradient Flows: In Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics. Birkhäuser Verlag (2008). MR2401600
  3. BALDASSARRI, A. - PUGLISI, A. - MARINI BETTOLO MARCONI, U., Kinetics models of inelastic gases. Math. Models Methods Appl. Sci., 12 (2002), 965-983. Zbl1174.82326MR1918169DOI10.1142/S0218202502001982
  4. BASSETTI, F. - LADELLI, L. - MATTHES, D., Central limit theorem for a class of one-dimensional kinetic equations. Probab. Theory Related Fields (2010) (Published on line). Zbl1225.82055MR2800905DOI10.1007/s00440-010-0269-8
  5. BASSETTI, F. - LADELLI, L. - REGAZZINI, E., Probabilistic study of the speed of approach to equilibrium for an inelastic Kac model. J. Stat. Phys., 133 (2008), 683-710 Zbl1161.82337MR2456941DOI10.1007/s10955-008-9630-z
  6. BASSETTI, F. - GABETTA, E. - REGAZZINI, E., On the depth of the trees in the McKean representation of Wilds sums. Transport Theory Statist. Phys., 36 (2007), 421-438. Zbl1183.82053MR2357202DOI10.1080/00411450701468217
  7. BASSETTI, F. - TOSCANI, G., Explicit equilibria in a kinetic model of gambling. Phys. Rev. E, 81 (2010), 66-115. MR2736281DOI10.1103/PhysRevE.81.066115
  8. BEN-AVRAHAM, D. - BEN-NAIM, E. - LINDENBERG, K. - ROSAS, A., Self-similarity in random collision processes. Phys. Rev. E, 68 (2003). 
  9. BOBYLEV, A. V., The theory of the spatially Uniform Boltzmann equation for Maxwell molecules. Sov. Sci. Review C, 7 (1988), 112-229. MR1128328
  10. BOBYLEV, A. V. - CARRILLO, J. A. - GAMBA, I. M., On some properties of kinetic and hydrodynamic equations for inelastic interactions. J. Statist. Phys., 98 (2000), 743-773. Zbl1056.76071MR1749231DOI10.1023/A:1018627625800
  11. BOBYLEV, A. V. - CERCIGNANI, C. - GAMBA, I. M., On the self-similar asymptotics for generalized nonlinear kinetic maxwell models. Comm. Math. Phys., 291 (2009), 599-644. Zbl1192.35126MR2534787DOI10.1007/s00220-009-0876-3
  12. BOBYLEV, A. V. - CERCIGNANI, C., Exact eternal solutions of the Boltzmann equation. J. Stat. Phys., 106 (2002), 1019-1038. Zbl1001.82090MR1889600DOI10.1023/A:1014085719973
  13. BOBYLEV, A. V. - CERCIGNANI, C., Self similar solutions of the Boltzmann equation and their applications. J. Stat. Phys., 106 (2002), 1039-1071. Zbl1001.82091MR1889601DOI10.1023/A:1014037804043
  14. BREIMAN, L., Probability. Corrected reprint of the 1968 original. Classics in Applied Mathematics, 7. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1992). MR1163370DOI10.1137/1.9781611971286
  15. CARLEN, E. A. - CARVALHO, M. C. - GABETTA, E., Central limit theorem for Maxwellian molecules and truncation of the Wild expansion. Comm. Pure Appl. Math., 53 (2000), 370-397. Zbl1028.82017MR1725612DOI10.1002/(SICI)1097-0312(200003)53:3<370::AID-CPA4>3.0.CO;2-0
  16. CARLEN, E. A. - CARVALHO, M. C. - GABETTA, E., On the relation between rates of relaxation and convergence of Wild sums for solutions of the Kac equation. J. Funct. An., 220 (2005), 362-387. Zbl1108.82036MR2119283DOI10.1016/j.jfa.2004.06.011
  17. CARLEN, E. A. - GABETTA, E. - TOSCANI, G., Propagation of smoothness and the rate of exponential convergence to equilibrium for a spatially homogeneous Maxwellian gas. Commun. Math. Phys., 199 (1999), 521-546. Zbl0927.76088MR1669689DOI10.1007/s002200050511
  18. CARLEN, E. A. - GABETTA, E. - REGAZZINI, E., On the rate of explosion for infinite energy solutions of the spatially homogeneous Boltzmann equation. J. Stat. Phys., 129 (2007), 699-723. Zbl1131.82023MR2360229DOI10.1007/s10955-007-9403-0
  19. CARLEN, E. A. - GABETTA, E. - REGAZZINI, E., Probabilistic investigations on the explosion of solutions of the Kac equation with infinite energy initial distribution. J. Appl. Probab., 45 (2008), 95-106. Zbl1142.60013MR2409313DOI10.1239/jap/1208358954
  20. CARLEN, E. A. - LU, X., Fast and slow convergence to equilibrium for Maxwellian molecules via Wild sums. J. Stat. Phys., 112 (2003), 59-134. Zbl1079.82012MR1991033DOI10.1023/A:1023623503092
  21. CERCIGNANI, C., Theory and application of the Boltzmann equation. Elsevier, New York (1975). Zbl0403.76065MR406273
  22. CERCIGNANI, C., The Boltzmann equation and its applications. Applied Mathematical Sciences, 67. Springer-Verlag, New York (1988). Zbl0646.76001MR1313028DOI10.1007/978-1-4612-1039-9
  23. CERCIGNANI, C., Mathematical methods in kinetic theory. Second edition. Plenum Press, New York (1990). Zbl0726.76083MR1069558DOI10.1007/978-1-4899-7291-0
  24. CERCIGNANI, C., Shear flow of a granular material. J. Statist. Phys., 102 (2001), 1407-1415. Zbl0990.82023MR1830452DOI10.1023/A:1004804815471
  25. CERCIGNANI, C. - ILLNER, R. - PULVIRENTI, M., The mathematical theory of dilute gases. Applied Mathematical Sciences, 106. Springer-Verlag, New York (1994). Zbl0813.76001MR1307620DOI10.1007/978-1-4419-8524-8
  26. C. CERCIGNANI - E. GABETTA Eds., Transport phenomena and kinetic theory. Applications to Gases, Semiconductors, Photons, and Biological Systems. Model. Simul. Sci. Eng. Technol., BirkhäuserBoston, MA (2007). MR2334302DOI10.1007/978-0-8176-4554-0
  27. CRAMÉR, H., On the approximation to a stable probability distribution. In Studies in Mathematical Analysis and Related Topics. Stanford Univ. Press. (1962), 70-76. MR146874
  28. CRAMÉR, H., On asymptotic expansions for sums of independent random variables with a limiting stable distribution. Sankhya Ser. A, 25 (1963), 13-24. Addendum, ibid. 216. MR174079
  29. DOLERA, E. - GABETTA, E. - REGAZZINI, E., Reaching the best possible rate of convergence to equilibrium for solutions of Kac's equation via central limit theorem. Ann. Appl. Probab., 19 (2009), 186-209. Zbl1163.60007MR2498676DOI10.1214/08-AAP538
  30. DOLERA, E. - REGAZZINI, E., The role of the central limit theorem in discovering sharp rates of convergence to equilibrium for the solution of the Kac equation. Ann. Appl. Probab., 20 (2010), 430-461. Zbl1195.60033MR2650038DOI10.1214/09-AAP623
  31. DRMOTA, M., Random trees. An interplay between combinatorics and probability. SpringerWienNew York, Vienna (2009). Zbl1170.05022MR2484382DOI10.1007/978-3-211-75357-6
  32. DURRETT, R. - LIGGETT, T. M., Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebiete, 64 (1983), 275-301. Zbl0506.60097MR716487DOI10.1007/BF00532962
  33. FORTINI, S. - LADELLI, L. - REGAZZINI, E., A central limit problem for partially exchangeable random variables. Theory Probab. Appl., 41 (1996), 224-246. Zbl0881.60019MR1445757DOI10.1137/S0040585X97975459
  34. GABETTA, E., Results on optimal rate of convergence to equilibrium for spatially homogeneous Maxwellian gases. In Transport phenomena and kinetic theory (2007), 19-37, Model. Simul. Sci. Eng. Technol., BirkhäuserBoston, MA. Zbl1121.82034MR2334304DOI10.1007/978-0-8176-4554-0_2
  35. GABETTA, E. - REGAZZINI, E., Some new results for McKean's graphs with applications to Kac's equation. J. Stat. Phys., 125 (2006), 947-974. Zbl1107.82046MR2283786DOI10.1007/s10955-006-9187-7
  36. GABETTA, E. - REGAZZINI, E., Central limit theorem for the solution of the Kac equation. Ann. Appl. Probab., 18 (2008), 2320-2336. Zbl1161.82018MR2474538DOI10.1214/08-AAP524
  37. GABETTA, E. - REGAZZINI, E., Central limit theorem for the solution of the Kac equation: Speed of approach to equilibrium in weak metrics. Probab Theory Related Fields, 146 (2010), 451-480. Zbl1181.60030MR2574735DOI10.1007/s00440-008-0196-0
  38. GNEDENKO, B. V. - KOLMOGOROV, A. N., Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA (1954). Zbl0056.36001MR62975
  39. GOLDIE, C. M. - GRUÈBEL, R., Perpetuities with thin tails. Adv. in Appl. Probab., 28 (1996), 463-480. Zbl0862.60046MR1387886DOI10.2307/1428067
  40. GOLDIE, C. M. - MALLER, R. A., Stability of perpetuities. Ann. Probab., 28 (2000), 1195-1218. Zbl1023.60037MR1797309DOI10.1214/aop/1019160331
  41. FISCHER, H., History of the Central Limit Theorem. Springer (2010). 
  42. HALL, P., Two-sided bounds on the rate of convergence to a stable law. Z. Wahrsch. Verw. Gebiete, 57 (1981), 349-364. Zbl0451.60026MR629531DOI10.1007/BF00534829
  43. IBRAGIMOV, I. A. - LINNIK, Y. V., Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publishing, Groningen (1971). Zbl0219.60027MR322926
  44. KAC, M., Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3 (1954-1955), 171-197. University of California Press, Berkeley and Los Angeles (1956). MR84985
  45. MCKEAN JR, H. P.., Speed of approach to equilibrium for Kac's caricature of a Maxwellian gas. Arch. Rational Mech. Anal., 21 (1966), 343-367. Zbl1302.60049MR214112DOI10.1007/BF00264463
  46. MCKEAN JR, H. P.., An exponential formula for solving Boltmann's equation for a Maxwellian gas. J. Combinatorial Theory, 2 (1967), 358-382. Zbl0152.46501MR224348
  47. LIU, Q., Fixed points of a generalized smoothing transformation and applications to the branching random walk. Adv. in Appl. Probab., 30 (1998), 85-112. Zbl0909.60075MR1618888DOI10.1239/aap/1035227993
  48. LIU, Q., On generalized multiplicative cascades. Stochastic Process. Appl., 86 (2000), 263-286. Zbl1028.60087MR1741808DOI10.1016/S0304-4149(99)00097-6
  49. MATTHES, D. - TOSCANI, G., On steady distributions of kinetic models of conservative economies. J. Stat. Phys., 130 (2008), 1087-1117. Zbl1138.91020MR2379241DOI10.1007/s10955-007-9462-2
  50. MATTHES, D. - TOSCANI, G., Propagation of Sobolev regularity for a class of random kinetic models on the real line. Nonlinearity, 23 (2010), 2081-2100 Zbl1203.82073MR2672637DOI10.1088/0951-7715/23/9/003
  51. PETROV, V. V., Limit theorems of probability theory. Sequences of independent random variables. Oxford Studies in Probability. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1995). Zbl0826.60001MR1353441
  52. PULVIRENTI, A. - TOSCANI, G., Asymptotic properties of the inelastic Kac model. J. Statist. Phys., 114 (2004), 1453-1480. Zbl1072.82030MR2039485DOI10.1023/B:JOSS.0000013964.98706.00
  53. REGAZZINI, E., Convergence to Equilibrium of the Solution of Kac's Kinetic Equation. A Probabilistic View. Bollettino UMI, 2 (2009), 175-198. Zbl1177.82093MR2493650
  54. RACHEV, S. T., Probability metrics and the stability of stochastic models. Wiley, New York (1991). Zbl0744.60004MR1105086
  55. SZNITMAN, A. S., Èquations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete, 66 (1986), 559-592. MR753814DOI10.1007/BF00531891
  56. TANAKA, H., An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas. Z. Wahrsch. Verw. Gebiete, 27 (1973), 47-52. Zbl0302.60005MR362442DOI10.1007/BF00736007
  57. TANAKA, H., Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete, 46 (1978), 67-105. Zbl0389.60079MR512334DOI10.1007/BF00535689
  58. TOSCANI, G., Wealth redistribution in conservative linear kinetic models with taxation. Europhysics Letters, 88 (2009), 10007. 
  59. VERVAAT, W., On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Probab., 11 (1979), 750-783. Zbl0417.60073MR544194DOI10.2307/1426858
  60. VILLANI, C., Topics in optimal transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence, RI (2003). Zbl1106.90001MR1964483DOI10.1007/b12016
  61. VILLANI, C., Optimal transport. Old and new. Grundlehren der Mathematischen Wissenschaften, 338. Springer-Verlag, Berlin (2009). Zbl1156.53003MR2459454DOI10.1007/978-3-540-71050-9
  62. WILD, E., On Boltzmann's equation in the kinetic theory of gases. Proc. Cambridge Philos. Soc., 47 (1951), 602-609. Zbl0043.43703MR42999

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.