Clustering of linearly interacting diffusions and universality of their long-time limit distribution

Let K subset of R-d (d greater than or equal to 1) be a compact convex set and Lambda a countable Abelian group. We study a stochastic process X in K- Lambda, equipped with the product topology, where each coordinate solves a SDE of the form dX(i)(t) = Sigma (j)a(j - i)(X-j(t) - X-i(t))dt + sigma (X- i(t))dB(i)(t). Here a(.) is the kernel of a continuous-time random walk on Lambda and sigma is a continuous root of a diffusion matrix omega on K. If X(t) converges in distribution to a limit X (infinity) and the symmetrized random walk with kernel a(S)(i) = a(i) + a(-i) is recurrent, then each comp onent X-i(infinity) is concentrated on {x is an element of K : sigma (x) = 0} and the coordinates agree, i.e,the system clusters. Both these statement s fail if a(S) is transient. Under the assumption that the class of harmoni c functions of the diffusion matrix w is preserved under linear transformat ions of ri, we show that the system clusters for all spatially ergodic init ial conditions and we determine the limit distribution of the components. T his distribution turns out to be universal in all recurrent kernels a(S) on Abelian groups Lambda.