We study some aspects of the relationship between the long time behaviour of sys- tems with a finite but large number of components and their idealizations with countably many components. The following class of models is considered in detail, which contains examples occuring in population growth and population genetic models.
Let z(t) = {zi(t),i E H d } be the solution of the system of stochastic differential equations r 1
J
We assume a ( i , j ) is an irreducible random walk kernel on Z d , Z is an interval, g : Z --t R+ satisfies certain regularity conditions, and { wi(t), i E Z d } is a family of standard, independent Brownian motions on R. z ( t ) is an infinite system of interacting diffusions. The corresponding finite systems are z N ( t ) = { z y ( t ) , i E AN}, which solve a similar system of equations, with AN = ( - N , N I d n H d , anda ( i , j ) replaced by a N ( i , j ) = c, a ( i , j + 2Nk).
In the case where iL(i,j) = g ( a ( i , j ) + a ( j , z ) ) is recurrent, we prove, for example, that for Z = [0,1], respectively, [0, 00), for all t N t 00 as N + 00, respectively, if the initial distributions satisfy E z N ( 0 ) 0 (and an additional regularity condition in the case Z = [0, 00)). Here Furthermore, we give, in a particular model arising in population genetics, a detailed analysis of how the size of the "0 or 1 clusters" in the finite and infinite system compare.
Finally some analytical aspects of the analysis in the case where & ( i , j ) is transient are treated here.
IS(zN(tnr)) * s , ,
is the constant configuration, and 6, is the unit point mass on 1991 Mathematics Subject Classification. Primary: 60K35. Keywords and phrases. Interacting diffusions, finite particle systems, clustering. P ( z ( t ) is an IE'-valued 11 112 -continuous function of t 2 0) = 1,