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**Relative entropy and mixing properties of interacting particle systems.**
*(English)*
Zbl 0884.60094

Let \((S_t)\) be a Markov semigroup on a compact metric space \(X\) that has the Feller property. An interesting problem consists in obtaining conditions which guarantee that all limit points of \(\mu S_t\), \(t\geq 0\) (\(\mu\) being some initial distribution), are invariant measures. The case \(X= F^{\mathbb{Z}^d}\) (\(F\) being a finite set) is studied where \(X\) is the state space of an interacting particle system, its states being given by the values \(\eta(x)\), \(x\in \mathbb{Z}^d\). The infinitesimal generator is
\[
\Omega f(\eta)= \sum_{T\subset\mathbb{Z}^d} \int_{F^T} c_T(d\xi, \eta)(f(\eta^\xi)- f(\eta)),
\]
where the summation runs over the finite subsets of \(\mathbb{Z}^d\). Here \(c_T(d\xi, \eta)\) describes the rates for Poisson events that change the current configuration \(\eta\) to a new configuration \(\eta^\xi\) that has been altered on the finite set \(T\subset\mathbb{Z}^d\) from \(\eta\) to \(\xi\). The particle system is said to have bounded flip rates if
\[
\sup_{x\in\mathbb{Z}^d} \Biggl\{\sum_{T\ni x} \sup_{\eta\in X} \int_{F^T} c_T(d\xi, \eta)\Biggr\}<\infty.
\]
The particle system is said to have finite range \(R\) if the following two conditions are satisfied:

(a) \(c_T(\cdot, \cdot)\equiv 0\) if \(\sup_{x,y\in T} |x-y|\geq R\).

(b) For every \(A\subset F^T\) the function \(c_T(A, \eta)\) does not depend on \(\eta(u)\) if \(\text{dist}(T, u)\geq R\).

The main result says: Let the particle system have bounded flip rates and finite range. Let \(\mu\) be some probability measure on \(X\) such that the limit \(\lim_{n\to\infty} \mu S_{t_n}= \nu\) exists (where \(t_n\to\infty\)). Then \(\nu\) is invariant.

(a) \(c_T(\cdot, \cdot)\equiv 0\) if \(\sup_{x,y\in T} |x-y|\geq R\).

(b) For every \(A\subset F^T\) the function \(c_T(A, \eta)\) does not depend on \(\eta(u)\) if \(\text{dist}(T, u)\geq R\).

The main result says: Let the particle system have bounded flip rates and finite range. Let \(\mu\) be some probability measure on \(X\) such that the limit \(\lim_{n\to\infty} \mu S_{t_n}= \nu\) exists (where \(t_n\to\infty\)). Then \(\nu\) is invariant.

Reviewer: Klaus Schürger (Bonn)