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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 35, Fasc. 1,
pages 143 - 159
 

CAPTIVITY OF MEAN-FIELD PARTICLE SYSTEMS AND THE RELATED EXIT PROBLEMS

Julian Tugaut

Abstract: 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 hydro- dynamical 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.

2000 AMS Mathematics Subject Classification: Primary: 82C22, 60F10; Secondary: 60J60, 60G10.

Keywords and phrases: Interacting particle system, propagation of chaos, exit time, nonconvexity, free energy, invariant probabilities.

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