Ergodic Theorems for Infinite Systems of Locally Interacting Diffusions

Let x(t) = ᵢ(t), i ∈ Zᵈ\ be the solution of the system of stochastic differential equations dxᵢ(t) = (∑jᵈa(i,j)xⱼ(t) - xᵢ(t)) dt + √2g (xᵢ(t)) dwᵢ(t), i ∈ Zᵈ. Here g: 0, 1 → R⁺ satisfies $g > 0$ on (0, 1), $g(0) = g(1) = 0, g$ is Lipschitz, $a(i,j)$ is an irreducible random walk kernel on Zᵈ and ᵢ(t), i ∈ Zᵈ\ is a family of standard, independent Brownian motions on R; x(t) is a Markov process on X = 0, 1Zᵈ. This class of processes was studied by Notohara and Shiga; the special case $g(v) = v(1 - v)$ has been studied extensively by Shiga. We show that the long term behavior of $x(t)$ depends only on â(i,j) = (a(i,j) + a(j, i))/2 and is universal for the entire class of g considered. If â(i,j) is transient, then there exists a family \ν_θ, θ ∈ 0, 1\ of extremal, translation invariant equilibria. Each ν_θ is mixing and has density θ = ∫ x₀ dν_θ. If â(i,j), is recurrent, then the set of extremal translation invariant equilibria consists of the point masses \δ₀, δ₁\. The process starting in a translation invariant, shift ergodic measure μ on X with ∫ x₀ dμ = θ converges weakly as t → ∞ to ν_θ if â(i,j) is transient, and to (1 - θ)δ₀ + θδ₁ if â(i,j) is recurrent. (Our results in the recurrent case remove a mild assumption on g imposed by Notohara and Shiga.) For the case â(i,j) transient we use methods developed for infinite particle systems by Liggett and Spitzer. For the case â(i,j), recurrent we use a duality comparison argument.