Markov functions (Q5905410)

Let \(X=(X_ t)\) be a time-homogeneous Markov process with state space \(E\), and call a mapping \(\varphi: E\to F\) a Markov function if the image process \(\varphi(X_ t)\) is Markovian in its own right. The problem of constructing Markov functions has been studied by a number of authors, notably \textit{J. W. Pitman} and \textit{L. C. G. Rogers} [Ann. Probab. 9, 573-582 (1981; Zbl 0466.60070)] and \textit{J. Glover} and \textit{J. Mitro} [ibid. 18, No. 2, 655-668 (1990; Zbl 0714.60060)]. Suppose that the transition semigroup \((P_ t)\) of \(X\) is ``intertwined'' with a second family of kernels \((Q_ t)\) (not necessarily a semigroup), in the sense that there is a (proper) kernel \(K\) with \(P_ tK=KQ_ t\) for all \(t>0\). Let \(q>0\) be a function on \(E\) such that \(Kq\leq 1\). The author shows that if the range of \(K\) (operating on the positive measurable functions on \(E\)) coincides with the \(\sigma\)-algebra generated by said range, then \(\varphi(x)=K(x,dy)q(y)\) defines a Markov function taking values in the space \(F\) of subprobability measures on \(E\). A ``potential theoretic'' form of this result is also proved. In addition, it is shown how such examples of Markov functions arise naturally from consideration of the symmetries of \(X\).