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Ergodic theorems for infinite systems of locally interacting diffusions. (English) Zbl 0806.60100

Summary: Let \(x(t) = \{x_ i(t)\), \(i \in \mathbb{Z}^ d\}\) be the solution of the system of stochastic differential equations \[ dx_ i (t) = \biggl( \sum_{j \in \mathbb{Z}^ d} a(i,j) x_ j (t) - x_ i (t) \biggr) dt + \sqrt{ 2g \bigl( x_ i (t) \bigr)} dw_ i (t), \quad i \in \mathbb{Z}^ d. \] Here \(g:[0,1] \to \mathbb{R}^ +\) 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 \(X = [0,1]^{\mathbb{Z}^ d}\). This class of processes was studied by M. Notohara and T. Shiga [J. Math. Biol. 10, 281- 294 (1980; Zbl 0463.92011)]; the special case \(g(v) = v(1 - v)\) has been studied extensively by T. Shiga [J. Math. Soc. Jap. 39, No. 1, 17- 25 (1987; Zbl 0625.60095)].
We show that the long term behavior of \(x(t)\) depends only on \(\widehat a(i,j) = (a(i,j) + a(j,i))/2\) and is universal for the entire class of \(g\) considered. If \(\widehat a(i,j)\) is transient, then there exists a family \(\{\nu_ \theta, \theta \in [0,1]\}\) of extremal, translation invariant equilibria. Each \(\nu_ \theta\) is mixing and has density \(\theta = \int x_ 0 d \nu_ \theta\). If \(\widehat a(i,j)\) 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 \(t \to \infty\) to \(\nu_ \theta\) if \(\widehat a(i,j)\) is transient, and to \((1 - \theta) \delta_ 0 + \theta \delta_ 1\) if \(\widehat a(i,j)\) is recurrent. (Our results in the recurrent case remove a mild assumption on \(g\) imposed by Notohara and Shiga.) For the case \(\widehat a(i,j)\) transient we use methods developed for infinite particle systems by T. M. Liggett and F. Spitzer [Z. Wahrscheinlichkeitstheorie Verw. Geb. 56, 443- 468 (1981; Zbl 0444.60096)]. For the case \(\widehat a(i,j)\) recurrent we use a duality comparison argument.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
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