Occupation time Markov property for countable Markov chains with respect to several states
We generalize the results of the Markov property for the occupation time process τ for homogeneousMarkovchain.Supposethatthe“present” concentratenotinonestatebutinseveral (for simplicity, in two). Let us consider a random walk on a polygon and let us take two of its vertices as a “present”. It is known that in that case the occupation time τ is not Markovian. On theotherhanditisobviousthatif oneof the “present” verticesisessential(i.e.itsremovalleads the lost of connectivity) then τ is Markovian. It is interesting is there a Markov property on less restrictive assumptions. Unfortunately the answer is no. For some graphs there are another natural ways of Markov property generalization. For example, let us consider an inﬁnite “ladder” and the occupation time process for each “rung”. It could be shown the even in that case the occupation time process isn’t Markovian. Therefore in some sense the Markov property for the occupation time determines by the “present”. Or more precisely by an amount of states in the “present”.