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A martingale approach to the law of large numbers for weakly interacting stochastic processes. (English) Zbl 0544.60097
Two types of interacting processes are studied, namely diffusion processes and jump processes. N particles move in \({\mathbb{R}}^ n\) according to one of these evolution laws, where the drift vector and the diffusion matrix resp. the jump intensities not only depend on the position of the corresponding particle, but also on the total configuration. This paper treats the Vlasov limit of weak long range interaction, i.e. as N tends to infinity, the force is rescaled by 1/N, while its range remains fixed. It is proved that under suitable assumptions the process of the empirical measures converges weakly to a deterministic probability measure-valued process. The proof uses certain martingales which arise in this situation.
Reviewer: M.Mürmann

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60G42 Martingales with discrete parameter
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