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Feller processes on nonlocally compact spaces. (English) Zbl 1108.60064

Consider a Feller discrete-time Markov process whose state space is a complete separable metric space \(X\), and let \(P\) denote its transition function. No assumption of compacity or local compacity is made on \(X\). Assume that \(\limsup_{n\rightarrow \infty }(\frac{1}{n}\sum_{i=1}^{n}P^{i}( x,O))>0\) for some \(x\in X\), where \(O\) is an arbitrary open neighborhood of some point \(z\in X\). It is proved that if \(P\) is equicontinuous, then the process admits an invariant probability measure. Stability properties of such processes are also established.

MSC:

60J05 Discrete-time Markov processes on general state spaces
37A30 Ergodic theorems, spectral theory, Markov operators
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