The dual view of Markov Processes I

**Abstract**

This is the first of two talks on a dualized view of Markov processes.
The first part will describe how to view a probabilistic transition
system as a transformer of functions rather than as a transformer of
probability distributions. A Markov process is normally viewed as a Markov
kernel *i.e.* a map from
*S* x Σ → [0,1] where *S* is a state space and
Σ is a σ-algebra on *S*.
These Markov kernels are morphisms in the
Kleisli category of the Giry monad. In recent work by Chaput, Danos,
Panangaden and Plotkin, Markov processes were reinterpreted as linear maps on
the space of positive *L*_{1} functions on *S*.
This is analogous to taking the predicate transformer view of Markov
processes. A number of dualities and isomorphisms emerge in this
picture. Most interestingly conditional expectation can be understood
functorially.

In the second part (by Panangaden) the theory of bisimulation and approximation will be presented from this dualized viewpoint.