The e-property of asymptotically stable Markov semigroups. (English) Zbl 1534.60099
Summary: The relations between asymptotic stability, the eventual e-property and the e-property of Markov semigroups, acting on measures defined on general (Polish metric) spaces, are studied. While much attention is usually paid to asymptotic stability (for years the e-property has only served as a tool to establish it), it should be noted that the e-property itself is also important as it, e.g., ensures that numerical errors in simulations are negligible. Here, it is shown that any asymptotically stable Markov-Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the eventual e-property. Moreover, we prove that any Markov-Feller semigroup, which is strongly stochastically continuous, and which possesses the eventual (Polish metric), also has the e-property. We also present an example highlighting that the assumption of strong stochastic continuity (given in terms of the supremum norm) cannot be relaxed to its weak form, involving pointwise convergence, unless a state space of a process corresponding to a Markov semigroup is a compact metric space.
MSC:
| 60J25 | Continuous-time Markov processes on general state spaces |
| 37A30 | Ergodic theorems, spectral theory, Markov operators |
| 46N30 | Applications of functional analysis in probability theory and statistics |
| 46E27 | Spaces of measures |
| 60B10 | Convergence of probability measures |
Keywords:
Markov semigroup; e-property; equicontinuity; asymptotic stability; stochastic continuity; bounded-Lipschitz distanceReferences:
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