Finite and infinite systems of interacting diffusions: Cluster formation and universality properties

We study some aspects of the relationship between the long time behaviour of systems 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 cursive Greek chi(t) = {cursive Greek chi_i(t), i ∈ ℤ^d} be the solution of the system of stochastic differential equations formula presented We assume a(i, j) is an irreducible random walk kernel on ℤ^d, I is an interval, g : I → double-struck R sign⁺ satisfies certain regularity conditions, and {w_i(t),i ∈ ℤ^d} is a family of standard, independent Brownian motions on double-struck R sign. cursive Greek chi(t) is an infinite system of interacting diffusions. The corresponding finite systems are cursive Greek chi^N (t) = {cursive Greek chi^N_0i(t), i ∈ A_N}, which solve a similar system of equations, with A_N = (-N, N]^d ∩ ℤ^d, and a(i, j) replaced by a^N(i, j) = Σ_k a(i, j + 2Nk). In the case where â(i, j) = 1/2(a(i, j) + a(j, i)) is recurrent, we prove, for example, that for I = [0,1], respectively, [0, ∞), for all t_N ↑ ∞ ∞ N → ∞, formula presented respectively, formula presented if the initial distributions satisfy Ecursive Greek chi^N_i(0) ≡ θ (and an additional regularity condition in the case I = [0, ∞)). Here p is the constant configuration, and δp is the unit point mass on p. 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 â(i, j) is transient are treated here.