On Markov intertwining relations and primal conditioning
The article investigates \textit{intertwinings} of Markov chains. These are a type of similarity relation from a dual Markov process towards a primal process, enabling the transfer of information between the two. They have been used, for example, by \textit{D. Aldous} and \textit{P. Diaconis} [Am. Math. Mon. 93, 333--348 (1986; Zbl 0603.60006)] to analyse the convergence to equilibrium of the top-to-random shuffle.\N\NThe goal of the current article is to investigate when it is possible, and how, to transform a given Markov intertwining relation when the primal Markov chain is conditioned to stay in a proper subset of its state space.\N\NLet \(P\) be a transition matrix on a finite state space \(V\), and denote its corresponding discrete-time Markov chain \(X = (X_n)_{n\ge0}\). The dual Markov chain \(Y = (Y_n)_{n\ge0}\) takes values in another finite state space \(W\), and its transition matrix is denoted \(Q\). An \textit{algebraic intertwining relation} holds from \(Q\) to \(P\) if there exists a Markov kernel \(\Lambda\) from \(W\) to \(V\), called the \textit{link}, satisfying \(Q \Lambda = \Lambda P\).\N\NA more precise intertwining relation is also considered, enabling coupling of \(X\) and \(Y\). Further assume that for any \(x,x' \in V\) with \(P(x,x') > 0\) there is a Markov kernel \(K_{x,x'}\) on \(W\) from this, define the Markov kernel \(K\) on \(V \times W\) by\N\[\NK\bigl( (x,y), (x',y') \bigr) := P(x,x') K_{x,x'}(y,y').\N\]\NA \textit{probabilistic intertwining relation} from \(Q\) to \(P\) holds if\N\[\N\textstyle \sum_{x \in V} \Lambda(y,x) K\bigl( (x,y), (x',y') \bigr) = Q(y,y') \Lambda(y',x').\N\]\NSumming over \(y' \in W\), the algebraic intertwining relation is deduced. Conversely, \textit{P. Diaconis} and \textit{J. A. Fill} [Ann. Probab. 18, No. 4, 1483--1522 (1990; Zbl 0723.60083)] showed that it is always possible to associate a probabilistic intertwining relation to an algebraic one, with the same \(\Lambda\).\N\NThe purpose of the paper is to investigate the behaviour of a Markov intertwining under the conditioning that the primal \(X\) stays in a proper subset of \(V\). Three examples are studied.\N\begin{itemize}\N\item[1.] The discrete version of the Pitman intertwining between Brownian motion and the usual random walk on \(\mathbb Z\) conditioned to stay in a finite segment (aka, the Bessel-3 process)\N\item[2.] The top-to-random shuffle, with the card initially at the bottom conditioned to stay in the top half of the deck after reaching it\N\item[3.] Absorbed birth-and-death chains conditioned not to be absorbed\N\end{itemize}
algebraic and probabilistic Markov intertwining relations
scientific article; zbMATH DE number 7900854