The e-property of asymptotically stable Markov semigroups
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909145464045568 |
|---|---|
| author | Kukulski, Ryszard Wojewódka-Ściążko, Hanna |
| author_facet | Kukulski, Ryszard Wojewódka-Ściążko, Hanna |
| contents | 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 usually much attention is paid to asymptotic stability (and the e-property has been for years verified only 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 e-property, also has the e-property. We also present an example highlighting that strong stochastic continuity cannot be replaced by its weak counterpart, unless a state space of a process corresponding to a Markov semigroup is a compact metric space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_16424 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The e-property of asymptotically stable Markov semigroups Kukulski, Ryszard Wojewódka-Ściążko, Hanna Probability 60J25, 37A30, 46N30, 46E27, 60B10 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 usually much attention is paid to asymptotic stability (and the e-property has been for years verified only 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 e-property, also has the e-property. We also present an example highlighting that strong stochastic continuity cannot be replaced by its weak counterpart, unless a state space of a process corresponding to a Markov semigroup is a compact metric space. |
| title | The e-property of asymptotically stable Markov semigroups |
| topic | Probability 60J25, 37A30, 46N30, 46E27, 60B10 |
| url | https://arxiv.org/abs/2211.16424 |