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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
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Existence of an invariant measure on an infinite dimensional space via Lyapunov functional
Is seems to me that your proof is correct. The same restriction to locally compact state spaces appears in the classical ''Markov Chains and Stochastic Stability'' by Meyne and Tweedie (chap. 12).
My …