The Annals of Probability

Propagation of chaos for the Landau equation with moderately soft potentials

Nicolas Fournier and Maxime Hauray

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the 3D Landau equation for moderately soft potentials [$\gamma\in(-2,0)$ with the usual notation] as well as a stochastic system of $N$ particles approximating it. We first establish some strong/weak stability estimates for the Landau equation, which are fully satisfactory only when $\gamma\in[-1,0)$. We next prove, under some appropriate conditions on the initial data, the so-called propagation of molecular chaos, that is, that the empirical measure of the particle system converges to the unique solution of the Landau equation. The main difficulty is the presence of a singularity in the equation. When $\gamma\in(-1,0)$, the strong-weak uniqueness estimate allows us to use a coupling argument and to obtain a rate of convergence. When $\gamma\in(-2,-1]$, we use the classical martingale method introduced by McKean. To control the singularity, we have to take advantage of the regularity provided by the entropy dissipation. Unfortunately, this dissipation is too weak for some (very rare) aligned configurations. We thus introduce a perturbed system with an additional noise, show the propagation of chaos for this system and finally prove that the additional noise is almost never used in the limit $N\to\infty$.

Article information

Ann. Probab., Volume 44, Number 6 (2016), 3581-3660.

Received: January 2015
Revised: July 2015
First available in Project Euclid: 14 November 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C40: Kinetic theory of gases
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 65C05: Monte Carlo methods

Landau equation uniqueness stochastic particle systems propagation of chaos Fisher information entropy dissipation


Fournier, Nicolas; Hauray, Maxime. Propagation of chaos for the Landau equation with moderately soft potentials. Ann. Probab. 44 (2016), no. 6, 3581--3660. doi:10.1214/15-AOP1056.

Export citation


  • [1] Arsen’ev, A. A. and Peskov, N. V. (1977). The existence of a generalized solution of Landau’s equation. Ž. Vyčisl. Mat. i Mat. Fiz. 17 1063–1068, 1096.
  • [2] Bobylev, A. V., Pulvirenti, M. and Saffirio, C. (2013). From particle systems to the Landau equation: A consistency result. Comm. Math. Phys. 319 683–702.
  • [3] Fournier, N. and Guillin, A. (2015). From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules. Preprint.
  • [4] Carrapatoso, K. (2016). Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. Kinet. Relat. Models 9 1–49.
  • [5] Cépa, E. and Lépingle, D. (2001). Brownian particles with electrostatic repulsion on the circle: Dyson’s model for unitary random matrices revisited. ESAIM Probab. Stat. 5 203–224 (electronic).
  • [6] de Finetti, B. (1937). La prévision : Ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincaré 7 1–68.
  • [7] Desvillettes, L. (2015). Entropy dissipation estimates for the Landau equation in the Coulomb case and applications. J. Funct. Anal. 269 1359–1403.
  • [8] Desvillettes, L. and Villani, C. (2000). On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25 179–259.
  • [9] Desvillettes, L. and Villani, C. (2000). On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications. Comm. Partial Differential Equations 25 261–298.
  • [10] Figalli, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254 109–153.
  • [11] Fontbona, J., Guérin, H. and Méléard, S. (2009). Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Probab. Theory Related Fields 143 329–351.
  • [12] Fournier, N. (2009). Particle approximation of some Landau equations. Kinet. Relat. Models 2 451–464.
  • [13] Fournier, N. (2010). Uniqueness of bounded solutions for the homogeneous Landau equation with a Coulomb potential. Comm. Math. Phys. 299 765–782.
  • [14] Fournier, N. and Guérin, H. (2009). Well-posedness of the spatially homogeneous Landau equation for soft potentials. J. Funct. Anal. 256 2542–2560.
  • [15] 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.
  • [16] Fournier, N., Hauray, M. and Mischler, S. (2014). Propagation of chaos for the 2D viscous vortex model. J. Eur. Math. Soc. (JEMS) 16 1423–1466.
  • [17] Funaki, T. (1985). The diffusion approximation of the spatially homogeneous Boltzmann equation. Duke Math. J. 52 1–23.
  • [18] Givens, C. R. and Shortt, R. M. (1984). A class of Wasserstein metrics for probability distributions. Michigan Math. J. 31 231–240.
  • [19] Gualdani, M. P. and Guillen, N. (2014). Estimates for radial solutions of the homogeneous Landau equation with Coulomb potential. Available at
  • [20] Hauray, M. and Jabin, P.-E. (2015). Particle approximation of Vlasov equations with singular forces: Propagation of chaos. Ann. Sci. Éc. Norm. Supér. (4) 48 891–940.
  • [21] Hauray, M. and Mischler, S. (2014). On Kac’s chaos and related problems. J. Funct. Anal. 266 6055–6157.
  • [22] Hewitt, E. and Savage, L. J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80 470–501.
  • [23] Kac, M. (1956). Foundations of kinetic theory. In Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 19541955, Vol. III 171–197. Univ. California Press, Berkeley and Los Angeles.
  • [24] Méléard, S. (1996). Asymptotic behaviour of some interacting particle systems; McKean–Vlasov and Boltzmann models. In Probabilistic Models for Nonlinear Partial Differential Equations (Montecatini Terme, 1995). Lecture Notes in Math. 1627 42–95. Springer, Berlin.
  • [25] Miot, E., Pulvirenti, M. and Saffirio, C. (2011). On the Kac model for the Landau equation. Kinet. Relat. Models 4 333–344.
  • [26] Mischler, S. and Mouhot, C. (2013). Kac’s program in kinetic theory. Invent. Math. 193 1–147.
  • [27] 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.
  • [28] Osada, H. (1986). Propagation of chaos for the two-dimensional Navier–Stokes equation. Proc. Japan Acad. Ser. A Math. Sci. 62 8–11.
  • [29] Osada, H. (1987). Propagation of chaos for the two-dimensional Navier–Stokes equation. In Probabilistic Methods in Mathematical Physics (Katata/Kyoto, 1985) 303–334. Academic Press, Boston, MA.
  • [30] Pardoux, E. and Răşcanu, A. (2014). Stochastic Differential Equations, Backward SDEs, Partial Differential Equations. Stochastic Modelling and Applied Probability 69. Springer, Cham.
  • [31] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Springer, Berlin.
  • [32] Robinson, D. W. and Ruelle, D. (1967). Mean entropy of states in classical statistical mechanics. Comm. Math. Phys. 5 288–300.
  • [33] Rosenthal, H. P. (1970). On the subspaces of $L_{p}(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8 273–303.
  • [34] Stroock, D. W. and Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften 233. Springer, Berlin.
  • [35] Sznitman, A.-S. (1984). Équations de type de Boltzmann, spatialement homogènes. Z. Wahrsch. Verw. Gebiete 66 559–592.
  • [36] 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.
  • [37] Tanaka, H. (1978/79). Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Z. Wahrsch. Verw. Gebiete 46 67–105.
  • [38] Villani, C. (1998). On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Ration. Mech. Anal. 143 273–307.
  • [39] 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.
  • [40] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.