  Many behavioural equivalences or preorders for probabilistic processes
involve a lifting operation that turns a relation on states into a relation on
distributions of states. We show that several existing proposals for lifting
relations can be reconciled to be different presentations of essentially the
same lifting operation. More interestingly, this lifting operation nicely
corresponds to the Kantorovich metric, a fundamental concept used in
mathematics to lift a metric on states to a metric on distributions of states,
besides the fact the lifting operation is related to the maximum flow problem
in optimisation theory.
  The lifting operation yields a neat notion of probabilistic bisimulation, for
which we provide logical, metric, and algorithmic characterisations.
Specifically, we extend the Hennessy-Milner logic and the modal mu-calculus
with a new modality, resulting in an adequate and an expressive logic for
probabilistic bisimilarity, respectively. The correspondence of the lifting
operation and the Kantorovich metric leads to a natural characterisation of
bisimulations as pseudometrics which are post-fixed points of a monotone
function. We also present an "on the fly" algorithm to check if two states in a
finitary system are related by probabilistic bisimilarity, exploiting the close
relationship between the lifting operation and the maximum flow problem.
