Ergodic Theorems for Infinite Systems of Locally Interacting Diffusions

Abstract

Let $x(t) = \{x_i(t), i \in \mathbb{Z}^d\}$ be the solution of the system of stochastic differential equations $dx_i(t) = \bigg(\sum_{j\in\mathbb{Z}^d}a(i,j)x_j(t) - x_i(t)\bigg) dt + \sqrt{2g (x_i(t))} dw_i(t), \quad i \in \mathbb{Z}^d.$ Here $g: \lbrack 0, 1\rbrack \rightarrow \mathbb{R}^+$ satisfies $g > 0$ on (0, 1), $g(0) = g(1) = 0, g$ is Lipschitz, $a(i,j)$ is an irreducible random walk kernel on $\mathbb{Z}^d$ and $\{w_i(t), i \in \mathbb{Z}^d\}$ is a family of standard, independent Brownian motions on $\mathbb{R}; x(t)$ is a Markov process on $X = \lbrack 0, 1\rbrack^{\mathbb{Z}^d}$. 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 $\hat{a}(i,j) = (a(i,j) + a(j, i))/2$ and is universal for the entire class of $g$ considered. If $\hat{a}(i,j)$ is transient, then there exists a family $\{\nu_\theta, \theta \in \lbrack 0, 1\rbrack\}$ of extremal, translation invariant equilibria. Each $\nu_\theta$ is mixing and has density $\theta = \int x_0 d\nu_\theta$. If $\hat{a}(i,j)$, is recurrent, then the set of extremal translation invariant equilibria consists of the point masses $\{\delta_0, \delta_1\}$. The process starting in a translation invariant, shift ergodic measure $\mu$ on $X$ with $\int x_0 d\mu = \theta$ converges weakly as $t \rightarrow \infty$ to $\nu_\theta$ if $\hat{a}(i,j)$ is transient, and to $(1 - \theta)\delta_0 + \theta\delta_1$ if $\hat{a}(i,j)$ is recurrent. (Our results in the recurrent case remove a mild assumption on $g$ imposed by Notohara and Shiga.) For the case $\hat{a}(i,j)$ transient we use methods developed for infinite particle systems by Liggett and Spitzer. For the case $\hat{a}(i,j)$, recurrent we use a duality comparison argument.