Symmetries and functions of Markov processes
The auhors discuss the question which functions of a Markov process yield a Markov process again. They consider a transient Hunt process X on an LCCB state space E. Let G denote the group of bimeasurable bijections of E onto itself that preserve the class \({\mathcal S}\) of excessive functions (i.e. \(\psi\in G\) iff \(\{f\circ \psi:\;f\in S\}={\mathcal S})\) and let H be a subgroup of G. They define F to be the quotient space of H-orbits and \(\phi\) the canonical projection of E onto F, and they set \(Y_ t:=\phi \circ X_ t\). Under the assumption that F is LCCB again (which in many applications is true, possibly after a slight modification) and that Y has no holding points and satisfies a certain transience condition, the main theorem states: There exists a continuous additive functional \((A_ t)\) of X with continuous inverse \((\tau_ t)\) such that \((Y_{\tau_ t},P^ x)\) and \((Y_{\tau_ t},P^{\psi (x)})\) are identical in law whenever \(\psi\in H\) and \((Y_{\tau_ t})\) is strong Markov with respect to \((P^ x)_{x\in E}\). In addition they discuss possible results in case the transience hypotheses on Y is dropped. In the meantime, there have been consecutive papers by the first author involving different symmetry groups and extending the above result [Math. Z. 204, No.1, 1-11 (1990; Zbl 0698.60065); Symmetry groups and translation invariant representations of Markov processes, Ann. Probab. (1990); Markov functions, Preprint (1990)].
scientific article; zbMATH DE number 4176151