  This paper investigates stochastic nondeterminism on continuous state spaces
by relating nondeterministic kernels and stochastic effectivity functions to
each other. Nondeterministic kernels are functions assigning each state a set o
subprobability measures, and effectivity functions assign to each state an
upper-closed set of subsets of measures. Both concepts are generalizations of
Markov kernels used for defining two different models: Nondeterministic
labelled Markov processes and stochastic game models, respectively. We show
that an effectivity function that maps into principal filters is given by an
image-countable nondeterministic kernel, and that image-finite kernels give
rise to effectivity functions. We define state bisimilarity for the latter,
considering its connection to morphisms. We provide a logical characterization
of bisimilarity in the finitary case. A generalization of congruences (event
bisimulations) to effectivity functions and its relation to the categorical
presentation of bisimulation are also studied.
