Constructing Strong Markov Processes

Markov processes that are not “strong Markov” can be made strongly Markov by enlarging the state space. In the classical theory, this is accomplished by assuming that the state space has a topological structure and then taking the compactification of that topological space. Such a process entails the unnatural assumption that the process lives in a topological space to start with. Here, we make no such assumption. Rather, we show how to introduce a natural topology associated with the process. We then complete the space with respect to this topology and show that the resulting Markov process has the strong Markov property in this enlarged space. Several examples illustrate the idea.