Any exchangeable Markov processes on $[k]^{ \mathbb{ N}}$ with cadlag sample paths projects to a Markov process on the simplex whose sample paths are cadlag and of locally bounded variation. Furthermore, any such process has a de Finetti-type description as a mixture of i.i.d. copies of time-inhomogeneous Markov processes on $[k]$. In the Feller case, these time-inhomogeneous Markov processes have a relatively simple structure; however, in the non-Feller case a greater variety of behaviors is possible since the transition law of the underlying Markov process on $[k]^{ \zz{ N}}$ can depend in a non-trivial way on the exchangeable $\sigma$-algebra of the process.
Categories: math.PR