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Captivity of mean-field particle systems and the related exit problems

Tugaut J.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

143--159

A mean-field system is a weakly interacting system of N particles in Rd confined by an external potential. The aim of this work is to establish a simple result about the exit problem of mean-field systems from some domains when the number of particles goes to infinity. More precisely, we prove the existence of some subsets of RdN such that the probability of leaving these sets before any T > 0 is arbitrarily small by taking N large enough. On the one hand, we show that the number of steady states in the small-noise limit is arbitrarily large with a sufficiently large number of particles. On the other hand, using the long-time convergence of the hydrodynamical limit, we identify the steady states as N goes to infinity with the invariant probabilities of the McKean-Vlasov diffusion so that some steady states in the small-noise limit are not steady states in the large N limit.

Interacting particle approximation for nonlocal quadratic evolution problems

Biler P., Funaki T., Woyczynski W. A.

Probability and Mathematical Statistics

1999

Vol. 19, Fasc. 2

267--286

The existence of McKean's nonlinear jump Markov processes and related Monte Carlo type approximation schemes by interacting particle systems (propagation of chaos) are studied for a class of multidimensional doubly nonlocal evolution problems with a fractional power of the Laplacian and a quadratic nonlinearity involving an integral operator. Asymptotically, these equations model the evolution of density of mutually interacting particles with anomalous (fractal) Lévy diffusion.