Topics in propagation of chaos. (English) Zbl 0732.60114

Sznitman, Alain-Sol

Calcul des probabilités, Ec. d’Été, Saint-Flour/Fr. 1989, Lect. Notes Math. 1464, 165-251 (1991).

[For the entire collection see Zbl 0722.00029.]
This article begins with the backgrounds of the probabilistic study for the propagation of chaos which goes back to M. Kac and H. P. McKean. The subjet is in some sense corresponding to the “mean field” method in the statistical physics and leads to some nonlinear PDE’s. The probabilistic approach is valuable since it provides the microscopic behavior of the systems not only the macroscopic behavior as provided by the PDE. As proved in the article, the stochastic processes studied in the subject are usually non-Markovian, but those treated in the article are closely linked with the Brownian motion. The author illustrates the main ideas by some typical models which lead to different types of PDE’s. In general, the processes are constructed from a sequence of scaled processes. This leads to study the existence and uniqueness of the limiting processes. Another topic of the study is on the asymptotic behavior of the marginal law which consists the essential part of the propagation of chaos. A large number of references are also included. Finally, refer to the reviewer’s book “From Markov chains to non-equilibrium particle systems” (World Scientific, 1991), Chapter 15 and Chapter 16, for some related study but from a different point of view.

Reviewer: Chen Mu-fa (Beijing)

MSC:

60K35

Interacting random processes; statistical mechanics type models; percolation theory

Keywords:

nonlinear processes; interacting particle systems; nonlinear partial differential equations; propagation of chaos; Brownian motion

Citations:

Zbl 0722.00029