On a Class of Set-Valued Markov Processes

Abstract

Let $Z$ be a finite or countable set, $\Xi$ the set of subsets of $Z, \{\xi_t\}$ a Markov process with state space $\Xi$. A process $\{\xi_t^\ast\}$ with the same state space is called associate to $\{\xi_t\}$ if $\mathbf{P}_\xi\{\xi_t \cap \eta \neq \varnothing\} = \mathbf{P}_\eta^\ast\{\xi_t^\ast \cap \xi \neq \varnothing\}$ whenever $\xi$ and $\eta$ are subsets of $\mathbf{Z}$, at least one of which is finite. Criteria are found for the existence of a process associate to a given one. Examples and applications are given.