A martingale approach to the law of large numbers for weakly
interacting stochastic processes
Ann. Probab., 12 (2):
458--479 (1984)Abstract
MR Consider a system of $N$ particles moving randomly in $R^n$. Two cases are studied. In one, the particles move according to diffusion processes, while in the other, they move by jumps. In both cases, the particles interact with one another. The interaction comes from letting the drift and diffusion coefficients in the first case, and the jump intensities in the second case, depend on the positions of all the particles. The positions of the $N$ particles are identified with the probability measure which puts mass $N^-1$ on each position. Thus, for each $N$, a certain measure-valued stochastic process is defined. The "laws of large numbers'' which are proved assert that, under certain conditions, these processes converge as $N\uparrowınfty$ to a deterministic measure-valued process. Roughly speaking, the assumptions are that the drift and diffusion coefficients in the first case and the jump intensities in the second case are Lipschitz continuous and do not grow too rapidly at infinity.