Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilitistic interpretation

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1. R. Alexandre et C. Villani, On the Landau approximation in plasma physics (in preparation).  
2. A.A. Arsenev and O.E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau–Fokker–Planck equation. Math. USSR Sbornik69 (1991) 465-478.  
3. K. Bichteler, J.B. Gravelreaux and J. Jacod, Malliavin calculus for processes with jumps, Theory and Application of stochastic Processes. Gordon and Breach, New York (1987).  
4. K. Bichteler and J. Jacod, Calcul de Malliavin pour les diffusions avec sauts, existence d'une densité pour le cas unidimensionel, in Séminaire de probabilités XVII. Springer, Berlin, Lecture Notes in Math. 986 (1983) 132-157.  
5. L. Boltzmann, Weitere studien über das wärme gleichgenicht unfer gasmoläkuler. Sitzungsber. Akad. Wiss.66 (1872) 275-370. Translation: Further Studies on the thermal equilibrium of gas molecules, S.G. Brush Ed., Pergamon, Oxford, Kinetic Theory2 (1966) 88-174.  
6. L. Boltzmann, Lectures on gas theory. Reprinted by Dover Publications (1995).  
7. P. Degon and B. Lucquin–Desreux, The Fokker-Planck asymptotics of the Boltzmann collision operator in the Coulomb case. Math. Mod. Meth. Appl. Sci.2 (1992) 167-182.  
8. L. Desvillettes, On asymptotics of the Boltzmann equation when the collisions become grazing. Transp. Theory Statist. Phys.21 (1992) 259-276.  
9. L. Desvillettes, C. Graham and S. Méléard, Probabilistic interpretation and numerical approximation of a Kac equation without cutoff. Stochastic Process. Appl.84 (1999) 115-135.  
10. N. Fournier, Existence and regularity study for two-dimensional Kac equation without cutoff by a probabilistic approach. Ann. Appl. Probab.10 (2000) 434-462.  
11. N. Fournier and S. Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comput.70 (2002) 583-604.  
12. T. Goudon, Sur l'équation de Boltzmann homogène et sa relation avec l'équation de Landau–Fokker–Planck : influence des collisions rasantes. C. R. Acad. Sci. Paris324 (1997) 265-270.  
13. C. Graham and S. Méléard, Existence and regularity of a solution of a Kac equation without cutoff using the stochastic calculus of variations. Comm. Math. Phys.205 (1999) 551-569.  
14. H. Guérin, Solving Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Probab.13 (2003) 515-539.  
15. H. Guérin, Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach. Stochastic Process. Appl.101 (2002) 303-325.  
16. H. Guérin and S. Méléard, Convergence from Boltzmann to Landau processes with soft potential and particle approximation. J. Statist. Phys.111 (2003) 931-966.  
17. J. Horowitz and R.L. Karandikar, Martingale problem associated with the Boltzmann equation, Seminar on Stochastic Processes, 1989, E. Cinlar, K.L. Chung and R.K. Getoor Eds., Birkhäuser, Boston (1990).  
18. J. Jacod and A.N. Shiryaev, Limit theorems for stochastic processes. Springer (1987).  
19. E.M. Lifchitz and L.P. Pitaevskii, Physical kinetics – Course in theorical physics. Pergamon, Oxford 10 (1981).  
20. D. Nualart, The Malliavin calculus and related topics. Springer-Verlag (1995).  
21. H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Geb.46 (1978) 67-105.  
22. C. Villani, On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Meth. Mod. Appl. Sci.8 (1998) 957-983.  
23. C. Villani, On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal.143 (1998) 273-307.