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Visible absorbing decompositions and uniqueness of invariant probabilities

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  • Jean-Gabriel Attali

Abstract

We identify the measurable absorbing obstruction to uniqueness of invariant probability measures for a Markov kernel. Ordinary absorbing decompositions obstruct global irreducibility and recurrence, but not necessarily uniqueness: an absorbing component may have full mass for no invariant probability. We prove that a Markov kernel has more than one invariant probability if and only if it admits a visible absorbing decomposition, namely two disjoint absorbing sets, each having full mass for an invariant probability. The proof uses only the Jordan decomposition of the difference of two invariant probabilities.

Suggested Citation

  • Jean-Gabriel Attali, 2026. "Visible absorbing decompositions and uniqueness of invariant probabilities," Papers 2601.04900, arXiv.org, revised May 2026.
    • Handle: RePEc:arx:papers:2601.04900
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    1. Douc, Randal & Fort, Gersende & Guillin, Arnaud, 2009. "Subgeometric rates of convergence of f-ergodic strong Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 897-923, March.
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