We consider Markov decision processes (MDP) as generators of sequences of probability distributions over states. A probability distribution is p-synchronizing if the probability mass is at least p in a single state, or in a given set of states. We consider four temporal synchronizing modes: a sequence of probability distributions is always p-synchronizing, eventually p-synchronizing, weakly p-synchronizing, or strongly p-synchronizing if, respectively, all, some, infinitely many, or all but finitely many distributions in the sequence are p-synchronizing. For each synchronizing mode, an MDP can be (i) sure winning if there is a strategy that produces a 1-synchronizing sequence; (ii) almost-sure winning if there is a strategy that produces a sequence that is, for all epsilon > 0, a (1-epsilon)-synchronizing sequence; (iii) limit-sure winning if for all epsilon > 0, there is a strategy that produces a (1-epsilon)-synchronizing sequence. We provide fundamental results on the expressiveness, decidability, and complexity of synchronizing properties for MDPs. For each synchronizing mode, we consider the problem of deciding whether an MDP is sure, almost-sure, or limit-sure winning, and we establish matching upper and lower complexity bounds of the problems: for all winning modes, we show that the problems are PSPACE-complete for eventually and weakly synchronizing, and PTIME-complete for always and strongly synchronizing. We establish the memory requirement for winning strategies, and we show that all winning modes coincide for always synchronizing, and that the almost-sure and limit-sure winning modes coincide for weakly and strongly synchronizing.
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