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arXiv · 2601.04900

Visible absorbing decompositions and uniqueness of invariant probabilities

Jean-Gabriel Attali
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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.

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Jean-Gabriel Attali. 2026-01-08. Visible absorbing decompositions and uniqueness of invariant probabilities. https://arxiv.org/abs/2601.04900

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