Visible absorbing decompositions and uniqueness of invariant probabilities

TL;DR

This paper characterizes when a Markov kernel has multiple invariant probabilities by identifying a visible absorbing decomposition as the key obstruction to uniqueness.

Contribution

It introduces the concept of visible absorbing decompositions and proves their equivalence to non-uniqueness of invariant probabilities using Jordan decomposition.

Findings

Multiple invariant probabilities occur if and only if a visible absorbing decomposition exists.

Absorbing components can have full mass without supporting an invariant probability.

The proof relies solely on Jordan decomposition of invariant probability differences.

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.