Rate of convergence of the nanbu particle system for hard potentials and maxwell molecules

Annals of Probability

Volumn 44, Issue 1, 2016, Pages 589-627

  Fournier, Nicolas ^a   Mischler, Stéphane ^b  

^a SORBONNE UNIVERSITÉ   (France)

^b UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

Kinetic theory; Propagation of chaos,wasserstein distance; Stochastic particle systems

Indexed keywords

EID: 84958225424     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/14-AOP983     Document Type: Article

References (44)

1. ALDOUS, D. (1978). Stopping times and tightness. Ann. Probab. 6 335-340.
2. ALEXANDRE, R. (2009). A review of Boltzmann equation with singular kernels. Kinet. Relat. Models. 2 551-646.
3. ALEXANDRE, R., DESVILLETTES, L., VILLANI, C. and WENNBERG, B. (2000). Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal. 152 327-355.
4. ARKERYD, L. (1972). On the Boltzmann equation. I. Existence. Arch. Ration. Mech. Anal. 45 1-16.
5. ARKERYD, L. (1972). On the Boltzmann equation. II. The full initial value problem. Arch. Ration. Mech. Anal. 45 17-34.
6. BARTHE, F. and BORDENAVE, C. (2013). Combinatorial optimization over two random point sets. In Séminaire de Probabilités XLV. Lecture Notes in Math. 2078 483-535. Springer, Cham.
7. BOBYLEV, A. V. (1997). Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems. J. Stat. Phys. 88 1183-1214.
8. BOISSARD, E. and LE GOUIC, T. (2014). On the mean speed of convergence of empirical and occupation measures in Wasserstein distance. Ann. Inst. Henri Poincaré Probab. Stat. 50 539-563.
9. BOLLEY, F., CAÑIZO, J. A. and CARRILLO, J. A. (2011). Stochastic mean-field limit:Non-Lipschitz forces and swarming. Math. Models Methods Appl. Sci. 21 2179-2210.
10. CARRAPATOSO, K. (2015). Quantitative and qualitative Kac's chaos on Boltzmann's sphere. Ann. Inst. Henri Poincaré Probab. 51 993-1039.
11. CERCIGNANI, C. (1988). The Boltzmann Equation and Its Applications. Applied Mathematical Sciences. 67. Springer, New York.
12. DEREICH, S., SCHEUTZOW, M. and SCHOTTSTEDT, R. (2013). Constructive quantization:Approximation by empirical measures. Ann. Inst. Henri Poincaré Probab. Stat. 49 1183-1203.
13. DESVILLETTES, L. (2001). Boltzmann's kernel and the spatially homogeneous Boltzmann equation. fluid dynamic processes with inelastic interactions at the molecular scale (Torino, 2000). Riv. Mat. Univ. Parma (6) 4* 1-22.
14. DESVILLETTES, L., GRAHAM, C. andMÉLÉARD, S. (1999). Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl. 84 115-135.
15. DESVILLETTES, L. andMOUHOT, C. (2009). Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions. Arch. Ration. Mech. Anal. 193 227-253.
16. ESCOBEDO, M. and MISCHLER, S. (2010). Scalings for a ballistic aggregation equation. J. Stat. Phys. 141 422-458.
17. FONTBONA, J., GUÉRIN, H. andMÉLÉARD, S. (2009). Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Probab. Theory Related Fields. 143 329-351.
18. FOURNIER, N. (2015). Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab. 25 860-897.
19. FOURNIER, N. and GODINHO, D. (2012). Asymptotic of grazing collisions and particle approximation for the Kac equation without cutoff. Comm. Math. Phys. 316 307-344.
20. FOURNIER, N. and GUÉRIN, H. (2008). On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity. J. Stat. Phys. 131 749-781.
21. FOURNIER, N. and GUILLIN, A. (2015). On the rate of convergence in Wasserstein distance of the empirical measure. Probab. Theory Related Fields. 162 707-738.
22. FOURNIER, N. and MÉLÉARD, S. (2002). A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comp. 71 583-604 (electronic).
23. FOURNIER, N. and MÉLÉARD, S. (2002). A weak criterion of absolute continuity for jump processes: Application to the Boltzmann equation. Bernoulli. 8 537-558.
24. FOURNIER, N. andMOUHOT, C. (2009). On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity. Comm. Math. Phys. 289 803-824.
25. GRAHAM, C. and MÉLÉARD, S. (1997). Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25 115-132.
26. GRÜNBAUM, F. A. (1971). Propagation of chaos for the Boltzmann equation. Arch. Ration. Mech. Anal. 42 323-345.
27. HAURAY, M. andMISCHLER, S. (2014). On Kac's chaos and related problems. J. Funct. Anal. 266 6055-6157.
28. KAC, M. (1956). Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, Vol. III 171-197. Univ. California Press, Berkeley.
29. LU, X. (1999). Conservation of energy, entropy identity, and local stability for the spatially homogeneous Boltzmann equation. J. Stat. Phys. 96 765-796.
30. LU, X. and MOUHOT, C. (2012). On measure solutions of the Boltzmann equation, part I:Moment production and stability estimates. J. Differential Equations. 252 3305-3363.
31. MCKEAN, H. P. (1975). Fluctuations in the kinetic theory of gases. Comm. Pure Appl. Math. 28 435-455.
32. MCKEAN, H. P. JR. (1967). An exponential formula for solving Boltmann's equation for a Maxwellian gas. J. Combinatorial Theory. 2 358-382.
33. MISCHLER, S. and MOUHOT, C. (2013). Kac's program in kinetic theory. Invent. Math. 193 1-147.
34. MISCHLER, S., MOUHOT, C. and WENNBERG, B. (2015). A new approach to quantitative propagation of chaos for drift, diffusion and jump processes. Probab. Theory Related Fields. 161 1-59.
35. MISCHLER, S. andWENNBERG, B. (1999). On the spatially homogeneous Boltzmann equation. Ann. Inst. H. Poincaré Anal. Non Linéaire. 16 467-501.
36. NANBU, K. (1983). Interrelations between various direct simulation methods for solving the Boltzmann equation. J. Phys. Soc. Japan. 52 3382-3388.
37. RACHEV, S. T. and RÜSCHENDORF, L. (1998). Mass Transportation Problems. Vol. I: Theory. Springer, New York.
38. SZNITMAN, A.-S. (1984). Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete. 66 559-592.
39. SZNITMAN, A.-S. (1991). Topics in propagation of chaos. In École d'Été de Probabilités de Saint-Flour XIX-1989. Lecture Notes in Math. 1464 165-251. Springer, Berlin.
40. TANAKA, H. (1978). On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. In Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) 409-425. Wiley, New York.
41. TANAKA, H. (1978/79). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete. 46 67-105.
42. TOSCANI, G. and VILLANI, C. (1999). Probability metrics and uniqueness of the solution to the Boltzmann equation for a Maxwell gas. J. Stat. Phys. 94 619-637.
43. VILLANI, C. (2002). A review of mathematical topics in collisional kinetic theory. In Handbook of Mathematical Fluid Dynamics, Vol. I 71-305. North-Holland, Amsterdam.
44. VILLANI, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics. 58. Amer. Math. Soc., Providence, RI.