Electronic Journal of Probability

The Landau equation for Maxwellian molecules and the Brownian motion on $SO_N(\mathbb{R})$

François Delarue, Stéphane Menozzi, and Eulalia Nualart

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In this paper we prove that the spatially homogeneous Landau equation for Maxwellian moleculescan be represented through the product of two elementary stochastic processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 92, 39 pp.

Accepted: 10 September 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60H40: White noise theory 60H10: Stochastic ordinary differential equations [See also 34F05]

Landau equation for Maxwellian molecules Stochastic analysis Heat kernel estimates on groups Large deviations

This work is licensed under aCreative Commons Attribution 3.0 License.


Delarue, François; Menozzi, Stéphane; Nualart, Eulalia. The Landau equation for Maxwellian molecules and the Brownian motion on $SO_N(\mathbb{R})$. Electron. J. Probab. 20 (2015), paper no. 92, 39 pp. doi:10.1214/EJP.v20-4012. https://projecteuclid.org/euclid.ejp/1465067198

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  • Aronson, D. G. Bounds for the fundamental solution of a parabolic equation. Bull. Amer. Math. Soc. 73 1967 890–896.
  • Bally, Vlad. On the connection between the Malliavin covariance matrix and Hörmander's condition. J. Funct. Anal. 96 (1991), no. 2, 219–255.
  • sc Carrapatoso, K. (2012) Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules. textit Preprint: http://arxiv.org/abs/1212.3724
  • scDelarue, F. (2012), Stochastic Analysis for the Complex Monge-Ampère Equation. (An Introduction to Krylov's Approach). Complex Monge-Ampère equations and Geodesics in the Space of Kähler Metrics. Lecture Notes in Mathematics, 2028, 55-198.
  • Desvillettes, Laurent; Villani, Cédric. On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness. Comm. Partial Differential Equations 25 (2000), no. 1-2, 179–259.
  • Fontbona, Joaquin; Guérin, Hélène; Méléard, Sylvie. Measurability of optimal transportation and convergence rate for Landau type interacting particle systems. Probab. Theory Related Fields 143 (2009), no. 3-4, 329–351.
  • Fournier, Nicolas. Particle approximation of some Landau equations. Kinet. Relat. Models 2 (2009), no. 3, 451–464.
  • Franchi, Jacques; Le Jan, Yves. Hyperbolic dynamics and Brownian motion. An introduction. Oxford Mathematical Monographs. Oxford University Press, Oxford, 2012. xvi+266 pp. ISBN: 978-0-19-965410-9.
  • Funaki, Tadahisa. The diffusion approximation of the Boltzmann equation of Maxwellian molecules. Publ. Res. Inst. Math. Sci. 19 (1983), no. 2, 841–886.
  • Funaki, Tadahisa. A certain class of diffusion processes associated with nonlinear parabolic equations. Z. Wahrsch. Verw. Gebiete 67 (1984), no. 3, 331–348.
  • Funaki, Tadahisa. The diffusion approximation of the spatially homogeneous Boltzmann equation. Duke Math. J. 52 (1985), no. 1, 1–23.
  • Funaki, Tadahisa. Construction of stochastic processes associated with the Boltzmann equation and its applications. Stochastic processes and their applications (Nagoya, 1985), 51–65, Lecture Notes in Math., 1203, Springer, Berlin, 1986.
  • Guérin, H. Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach. Stochastic Process. Appl. 101 (2002), no. 2, 303–325.
  • Guérin, Hélène. Solving Landau equation for some soft potentials through a probabilistic approach. Ann. Appl. Probab. 13 (2003), no. 2, 515–539.
  • Guérin, Hélène. Pointwise convergence of Boltzmann solutions for grazing collisions in a Maxwell gas via a probabilistic interpretation. ESAIM Probab. Stat. 8 (2004), 36–55 (electronic).
  • Guérin, Hélène; Méléard, Sylvie; Nualart, Eulalia. Estimates for the density of a nonlinear Landau process. J. Funct. Anal. 238 (2006), no. 2, 649–677.
  • Kusuoka, S.; Stroock, D. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985), no. 1, 1–76.
  • Kusuoka, S.; Stroock, D. Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 391–442.
  • Revuz, Daniel; Yor, Marc. Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 293. Springer-Verlag, Berlin, 1999. xiv+602 pp. ISBN: 3-540-64325-7.
  • sc Rogers, L.C.G. and Williams D. (1985) Markov processes and Martingales, Volume II: Itô Calculus. Cambridge University Press.
  • Stroock, Daniel W. Estimates for transition probabilities on a compact manifold. J. Funct. Anal. 242 (2007), no. 1, 295–303.
  • Sznitman, Alain-Sol. Topics in propagation of chaos. École d'Été de Probabilités de Saint-Flour XIX - 1989, 165–251, Lecture Notes in Math., 1464, Springer, Berlin, 1991.
  • Varopoulos, N. Th.; Saloff-Coste, L.; Coulhon, T. Analysis and geometry on groups. Cambridge Tracts in Mathematics, 100. Cambridge University Press, Cambridge, 1992. xii+156 pp. ISBN: 0-521-35382-3.
  • Villani, Cédric. On a new class of weak solutions to the spatially homogeneous Boltzmann and Landau equations. Arch. Rational Mech. Anal. 143 (1998), no. 3, 273–307.
  • Villani, C. On the spatially homogeneous Landau equation for Maxwellian molecules. Math. Models Methods Appl. Sci. 8 (1998), no. 6, 957–983.