A classification of bisimilarities for general Markov decision processes

We provide a fine classification of bisimilarities between states of possibly different labelled Markov processes (LMP). We show that a bisimilarity relation proposed by Panangaden that uses direct sums coincides with "event bisimilarity" from his joint work with Danos, Desharnais, and Laviolette. We also extend Giorgio Bacci's notions of bisimilarity between two different processes to the case of nondeterministic LMP and generalize the game characterization of state bisimilarity by Clerc et al. for the latter.
Comment: 40 pages. v2: Reference to prior example due to D. Gburek and added acknowledgment. v3: We thank Referee 1 for a simplified Example 3.32, and for making us realize that the argument involving Bernstein sets was flawed; Referee 2's observations helped us to streamline the discussion at the end of Section 3.4