LAWS OF LARGE NUMBERS FOR PAIRWISE INTERACTING PARTICLE SYSTEMS

We consider a system of Markov processes of finitely-many particles which exchange their energies in pairs at random times. A law of large numbers for this system means that the empirical measures of the processes may be approximated (as the number of particles increases) by the solution of a nonlinear evolution equation (the so-called McKean-Vlasov limit). This work presents two results of this type. The first one concerns the empirical processes and gives a probabilistic method for solving the nonlinear equation. The second is stated in the path scheme and extends classical results of chaos propagation by Kac (1956) and McKean (1967).