FINITE AND INFINITE SYSTEMS OF INTERACTING DIFFUSIONS

We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components. We consider in det ail a class of linearly interacting diffusions x(t) = {x(i)(t), i is a n element of Z(d)}, in the regime where there is a one-parameter famil y of nontrivial invariant measures. For these systems there are natura lly defined corresponding finite systems, x(N)(t) = {x(i)(N)(t), i is an element of Lambda(N)}, with Lambda(N) = (-N,N](d) boolean AND Z(d). Our main result gives a comparison between the laws of x(t(N)) and x( N)(t(N)) for times t(N) --> infinity as N --> infinity. The comparison involves certain mixtures of the invariant measures for the infinite system.