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The e-property of asymptotically stable Markov semigroups

Ryszard Kukulski, Hanna Wojewódka-Ściążko

Abstract

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.

The e-property of asymptotically stable Markov semigroups

Abstract

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.
Paper Structure (8 sections, 11 theorems, 54 equations)

This paper contains 8 sections, 11 theorems, 54 equations.

Table of Contents

  1. Preliminaries
  2. Spaces of Measures
  3. Markov Semigroups
  4. Ergodic Properties of Markov Semigroups
  5. Main Results
  6. Theorems
  7. Examples
  8. Proofs

Key Result

Theorem 2.1

Let $\{P(t)\}_{t\in\mathbb{R}_+}$ be an asymptotically stable Markov-Feller semigroup, and let $\mu_*$ be its unique invariant probability measure. If then $\{P(t)\}_{t\in\mathbb{R}_+}$ has the eventual e-property in $\mathcal{C}_b(S)$.

Theorems & Definitions (25)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Example 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • proof
  • ...and 15 more