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 detail a class of lincarly interacting diffusions x(t)={x _i (t), i ∈ ℤ ^d } in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally defined corresponding finite systems, {Mathematical expression}, with {Mathematical expression}. Our main result gives a comparison between the laws of x(t _N ) and x ^N (t _N ) for times t _N →∞ as N→∞. The comparison involves certain mixtures of the invariant measures for the infinite system.