Stone duality for Markov Processes

**Abstract:**

We define Aumann algebras, an algebraic analog of probabilistic modal
logic. An Aumann algebra consists of a Boolean algebra with operators
modeling probabilistic transitions. We prove that countable Aumann
algebras and countably-generated continuous-space Markov processes
defined on a certain class of Hausdorff spaces are dual in the sense
of Stone-type duality. The first idea might be to define Markov processes
on top of Stone spaces and just leverage ordinary Stone duality.
However, there are some twists in the tale. One has to remove
certain bad ultrafilters in order to get the duality. In doing so, one
loses compactness and one has to use other ideas. Our results subsume
existing results on completeness of probabilistic modal logics for Markov
processes. This is joint work with Dexter Kozen, Kim Larsen and Radu
Mardare.