We establish a Stone-type duality between specifications and infLMPs.
An infLMP is a probabilistic process whose transitions satisfy super-additivity
instead of additivity. Interestingly,  its simple structure can encode a mix of
probabilistic and non-deterministic behaviors.

We take the view of analysing probabilistic transition systems through events, or sets of states, as is usually done in probability theory.
It is well known that bisimulation for probabilistic processes without non determinism is characterized by a simple logic : two states are bisimilar if and only if they satisfy exactly the same formulas of that logic. Since formulas can be seen as sets of states, they  are ideal candidates for events: any LMP  can be associated to a morphism from the logical formulas to its sigma-algebra of states,  where the image of a formula  is the set of states that satisfy it.
We are interested in the converse: can a probabilistic process be defined by the set of its logical properties only? We can abstract the sigma-algebra of states as a $\sigma$-complete Boolean algebra and we can ask the question: when is it the case that a given function $\hat\mu$ mapping propositions of the logic to elements of an arbitrary ($\sigma$-complete) Boolean algebra  $A$ correspond to some LMP whose $\sigma$-algebra is isomorphic to $A$ and whose semantics accords with $\hat\mu$?
  We give a response to this question. opening the way to working with probabilistic processes in an abstract way, that is, without any explicit mention of the state space, manipulating properties only.  In other words, this opens the way to a Stone-type duality theory for these processes.


Our duality shows that an
infLMP can be considered as a demonic representative of a system's information.
Moreover, it carries forward a view where states are less important, and events,
or properties, become the main characters, as it should be in probability theory.
Along the way, we show that bisimulation and simulation are naturally interpreted
in this setting, and we exhibit the interesting relationship between infLMPs and the
usual probabilistic modal logics.

Most of this work  has been published in Concur 2009.