Kontoyiannis, I and Meyn, SP (2016) *On the f-norm ergodicity of markov processes in continuous time.* Electronic Communications in Probability, 21.

## Abstract

© 2016, University of Washington. All rights reserved. Consider a Markov process Φ = { Φ (t): t ≥ 0} evolving on a Polish space X. A version of the f -Norm Ergodic Theorem is obtained: Suppose that the process is ψ-irreducible and aperiodic. For a given function f: X → [1, ∞), under suitable conditions on the process the following are equivalent: ∫(i) There is a unique invariant probability measure π satisfying f dπ < ∞. (ii) There is a closed set C satisfying ψ(C) > 0 that is “self f -regular.” (iii) There is a function V: X → (0, ∞] that is finite on at least one point in X, for which the following Lyapunov drift condition is satisfied, (Formula Presented), (V3) where C is a closed small set and D is the extended generator of the process. For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of Φ under a suitable norm is also obtained: For each initial condition x ∈ X satisfying V (x) < ∞, and any function g: X → R for which |g| is bounded by f, (Formula Presented) Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process Φ or on the function g.

Item Type: | Article |
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Subjects: | UNSPECIFIED |

Divisions: | Div F > Signal Processing and Communications |

Depositing User: | Cron Job |

Date Deposited: | 08 Jan 2018 20:11 |

Last Modified: | 27 Oct 2020 07:53 |

DOI: | 10.1214/16-ECP4737 |