A family of Markov operators $P_i \colon C \to C, i \in I$ is given. Let $V_i$ be the set of the $P_i$-invariant measures. Is there any result in the literature about a necessary and sufficient condition (in terms of e.g. the compositions of $P_i$s) of $\bigcap_{i \in I} V_i \neq \emptyset$? In other words, when does it hold that a family of Markov operators have a common invariant measure?
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3$\begingroup$ I think without additional assumptions your question is far to general. Think of the similar question: Let $C$ be a topological/measurable space and $P_i \colon C \to C$, $i \in I$, an arbitrary family of continuous/measurable $\ldots$ mappings. When do they have a common fixpoint? $\endgroup$– Dieter KadelkaCommented Dec 3, 2020 at 22:55
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$\begingroup$ Primarily, I look for similar results in the literature, I am searching references. The problem relates to Markov chains. At first, one can assume that C is \mathbb{R}^n. $\endgroup$– Miklos PinterCommented Dec 5, 2020 at 8:33
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$\begingroup$ Could you specify what the space $C$ is (in case that it is not $\mathbb{R}^n$ as in your preceding comment)? $\endgroup$– Jochen GlueckCommented Dec 29, 2020 at 22:36
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1 Answer
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A sufficient condition: Commuting Markov operators will have a common invariant measure by https://en.wikipedia.org/wiki/Markov%E2%80%93Kakutani_fixed-point_theorem . See https://projecteuclid.org/download/pdf_1/euclid.kmj/1138845984 for some extensions
answered Dec 29, 2020 at 18:32