A Propagation of Chaos Result for Burgers’ Equation

  • A. S. Sznitman
Part of the The IMA Volumes in Mathematics and Its Applications book series (IMA, volume 9)


In 1967, McKean [1] conjectured that if one considered a system of N particles on R with formal generator
$$ {{L}_{N}}=\frac{1}{2}{{\sum\limits_{1}^{N}{\frac{\partial }{\partial x_{i}^{2}}}}^{2}}+\frac{1}{2(N-1)}\sum\limits_{i<j}{\delta (}{{x}_{i}}-{{x}_{j}})(\frac{\partial }{\partial {{x}_{i}}}+\frac{\partial }{\partial {{x}_{j}}}), $$
this system would present a propagation of chaos property with respect to Burgers’ equation:
$$ \frac{\partial u}{\partial t}=\frac{1}{2}\frac{{{\partial }^{2}}u}{\partial {{x}^{2}}}-\frac{1}{2}\frac{\partial }{\partial x}({{u}^{2}}), $$
that is to say, starting with N particles at time 0, independent and u0 distributed, if one fixes k, t > 0, when N goes to infinity, asymptotically the first k particles x1 t,...,xk t become independent and ut distributed, if ut is the solution of (1.2) with initial condition u0. More recently there has been several results in this direction, by CALDERONI-PULVIRENTI [1], OELSCHLAGER [1], introducing a smoothing ∅N(•) of δ(•), converging not too fast, and results by GUTKIN-KAC [1], KOTANI-OSADA [1], using analytical methods.


Brownian Motion Local Time Distribution Sense Discrete Time Markov Chain Uniqueness Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • A. S. Sznitman
    • 1
  1. 1.Laboratoire de ProbabilitésUniversite Paris VIParis, Cedex 05France

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