Symmetry groups of Markov processes and the diagonal principle (Q1174912)

The author continues his study of the symmetries of Markov processes, the focus of the present paper being symmetries of the cone of excessive measures. Let \(X=(X_ t,P^ x)\) be a transient right process with state space \(E\). Let \(\varphi: E\to E\) be a bijection which, along with its inverse, is measurable and finely continuous. Call \(\varphi\) a symmetry (of the cone of excessive measures of \(X\)) provided the image measure \(\varphi(\xi)\) is excessive if and only if \(\xi\) is excessive. The class Sym of such symmetries is a group under composition. Each subgroup \(H\) of Sym induces an orbit structure on \(E\) and a concomitant quotient map \(\Psi: E\to F\) (= the space of \(H\)-orbits). The author shows that the orbit-valued process \(\Psi(Z_ t)\) is a time-homogeneous Markov process; here \(Z\) is a process obtained by ``adjusting'' \(X\) through time change and \(h\)-transform. In the same vein, if \(H\) is transitive and such that the identity is the only element of \(H\) fixing a given base point \(e\in E\), then subject to certain side conditions, an adjusted form of \(X\) is \(H\)-translation invariant. The author conludes by suggesting a general framework for the study of symmetries of Markov processes, and illustrates the scheme through the example of the ``multiplicative'' symmetries of a regular step process.