Feller processes on nonlocally compact spaces

We consider Feller processes on a complete separable metric space X satisfying the ergodic condition of the form limsup n ->infinity (1/n Sigma P-n(i=1)i(x, 0)) > 0 for some x epsilon X, where O is an arbitrary open neighborhood of some point z epsilon X and P is a transition function. It is shown that e-chains which satisfy the above condition admit an invariant probability measure. Some results on the stability of such processes are also presented.