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Markov Chains Read chapter Read first chapter

2022 | OriginalPaper | Chapter

6. Petite Sets and Doeblin Points

Authors : Michel Benaïm, Tobias Hurth

Published in: Markov Chains on Metric Spaces

Publisher: Springer International Publishing

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Abstract

The notions of petite set, small set, and (weak) Doeblin point are discussed. For Feller chains, it is shown that the existence of an accessible weak Doeblin point implies irreducibility and thus unique ergodicity. This observation is applied to several examples in discrete and continuous time. In particular, for piecewise deterministic Markov processes and stochastic differential equations, it is shown how the accessibility condition can be phrased as a control problem and how Doeblin points are closely related to the classical Hörmander conditions for hypoelliptic diffusions.

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previous chapter Irreducibility
next chapter Harris and Positive Recurrence
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Metadata
Title
Petite Sets and Doeblin Points
Authors
Michel Benaïm
Tobias Hurth
Copyright Year
2022
Book
Markov Chains on Metric Spaces

Print ISBN: 978-3-031-11821-0

Electronic ISBN: 978-3-031-11822-7

Copyright Year: 2022

DOI
https://doi.org/10.1007/978-3-031-11822-7_6