Visible absorbing decompositions and uniqueness of invariant probabilities

Jean-Gabriel Attali

Papers from arXiv.org

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.

Date: 2026-01, Revised 2026-05
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