Bisimilarity of Distributionally Equivalent Markov Transition Systems

Markov transition systems for interpreting a simple negation free Hennessy-Milner logic are called distributionally equivalent iff for each formula the probability for its extension in one model is matched probabilistically in the other one. This extends in a natural way the notion of logical equivalence which is defined on the states of a transition system to its subprobability distributions. It is known that logical equivalence is equivalent to bisimilarity, i.e., the existence of a span of Borel maps that act as morphisms. We show that distributional equivalence is equivalent to bisimilarity as well, using a characterization of distributional equivalent transition systems through ergodic morphisms. As an aside, we relate bisimilar transition systems to those systems, for which cospans — taken in the category of measurable maps resp. in the Kleisli category associated with the Giry monad — exist.