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Uniqueness of invariant measures as a structural property of markov kernels

Jean-Gabriel Attali

Abstract

We identify indecomposability as a key measure-theoretic underlying uniqueness of invariant probability measures for discrete-time Markov kernels on general state spaces. The argument relies on the mutual singularity of distinct invariant ergodic measures and on the observation that uniqueness follows whenever all invariant probability measures are forced to charge a common reference measure. Once existence of invariant probability measures is known, indecomposability alone is sufficient to rule out multiplicity. On standard Borel spaces, this viewpoint is consistent with the classical theory: irreducibility appears as a convenient sufficient condition ensuring indecomposability, rather than as a structural requirement for uniqueness. The resulting proofs are purely measure-theoretic and do not rely on recurrence, regeneration, return-time estimates, or regularity assumptions on the transition kernel.

Uniqueness of invariant measures as a structural property of markov kernels

Abstract

We identify indecomposability as a key measure-theoretic underlying uniqueness of invariant probability measures for discrete-time Markov kernels on general state spaces. The argument relies on the mutual singularity of distinct invariant ergodic measures and on the observation that uniqueness follows whenever all invariant probability measures are forced to charge a common reference measure. Once existence of invariant probability measures is known, indecomposability alone is sufficient to rule out multiplicity. On standard Borel spaces, this viewpoint is consistent with the classical theory: irreducibility appears as a convenient sufficient condition ensuring indecomposability, rather than as a structural requirement for uniqueness. The resulting proofs are purely measure-theoretic and do not rely on recurrence, regeneration, return-time estimates, or regularity assumptions on the transition kernel.
Paper Structure (9 sections, 7 theorems, 45 equations)

This paper contains 9 sections, 7 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Quasi--Feller regularity
  4. Almost sure tightness of trajectories
  5. Stability via uniqueness
  6. From Duflo stability to positive Harris recurrence
  7. Harris-recurrence of Essential Feller Transition Kernel
  8. Further remarks
  9. Conclusion

Key Result

Lemma 3

Let $(E,\mathcal{B})$ be a standard Borel space and let $P$ be a Markov kernel on $(E,\mathcal{B})$. If $P$ admits two distinct invariant probability measures, then it admits two invariant probability measures that are mutually singular.

Theorems & Definitions (30)

  • Definition 1: Indecomposability
  • Remark 2
  • Lemma 3
  • proof
  • Remark 4
  • Proposition 5
  • proof
  • Remark 6
  • Corollary 7
  • proof
  • ...and 20 more