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The stationary behaviour of fluid limits of reversible processes is concentrated on stationary points. (English) Zbl 1278.60143

Suppose that \((Y^N)\) is a sequence of càdlàg stochastic processes on \([0,\infty)\) with values in a Polish space \(E\). Assuming existence of a deterministic continuous one-sided time flow \(\varphi\) on \(E\) such that \(Y^N\) converges to \(\varphi\) (the “fluid limit”) in a suitable sense, the following is proved: If \(Y^N\) is reversible with respect to some (then necessarily invariant) probability measure \(\pi^N\) on \(E\) for every \(N\), and if \(\pi\) is an accumulation point (in the topology of weak convergence) of the sequence \((\pi^N)\), then \(\varphi\) is reversible with respect to \(\pi\). Furthermore, \(\pi\) is concentrated on the set of fixed points of the fluid limit \(\varphi\).

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
37A60 Dynamical aspects of statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
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