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Criteria for asymptotic stability of eventually continuous Markov-Feller semigroups

Ting Li, Xianming Liu

Abstract

In this paper, we establish three criteria for the asymptotic behavior of Markov-Feller semigroups. First, we present a criterion for convergence in total variation to a unique invariant measure, requiring only $TV$-eventual continuity of the semigroup at a single point. Second, we propose two new criteria for asymptotic stability that require eventual continuity at a single point. This localized condition is more practical and easier to check. To illustrate the advantages of our framework, we provide an explicit example where verifying eventual continuity at a single point is straightforward, whereas establishing the corresponding global property is challenging.

Criteria for asymptotic stability of eventually continuous Markov-Feller semigroups

Abstract

In this paper, we establish three criteria for the asymptotic behavior of Markov-Feller semigroups. First, we present a criterion for convergence in total variation to a unique invariant measure, requiring only -eventual continuity of the semigroup at a single point. Second, we propose two new criteria for asymptotic stability that require eventual continuity at a single point. This localized condition is more practical and easier to check. To illustrate the advantages of our framework, we provide an explicit example where verifying eventual continuity at a single point is straightforward, whereas establishing the corresponding global property is challenging.

Paper Structure

This paper contains 10 sections, 6 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Notations and main results
  3. Application
  4. 2D Navier-Stokes equations with degenerate Lévy noise revisited
  5. Stability analysis for SDEs with multiplicative Poisson noise
  6. Proofs
  7. Proof of Theorem \ref{['thm0']}
  8. Proof of Theorem \ref{['thm2']}
  9. Proof of Corollary \ref{['col1']}
  10. Appendix

Key Result

Theorem 2.1

Let $\{P_t\}_{t\geq 0}$ be a Markov-Feller semigroup on $E$. Then the following two statements are equivalent: (1) $\{P_t\}_{t\geq 0}$ has a unique invariant measure $\mu$ and satisfies for any $\nu\in \mathcal{M}(E)$, (2) There exists $z\in E$ such that $\{P_t\}_{t\geq 0}$ is $TV$-eventually continuous at $z$ and for all $\epsilon >0$

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Proposition 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Proposition 3.1
  • proof
  • ...and 5 more