Jm. Swart, Clustering of linearly interacting diffusions and universality of their long-time limit distribution, PROB TH REL, 118(4), 2000, pp. 574-594
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.