Trajectories of stochastic systems from the point of view of categorical systems theory [efr-SKTW]

In David Jaz Myers' Categorical Systems Theory, we can form a doctrine of stochastic dynamical systems by (for example) choosing our category of spaces to be \mathsf {Set} and the category of bundles \mathcal {A}(X) to be the category Kl(\Delta )^X of families of (discrete) stochastic maps indexed by X. With TX defined as the constant family at X. Then a system is what you would call a discrete stochastic dynamical system. (With a technology for stochastic lenses, we could allow the base-level maps to be stochastic as well, but this is not of major importance here).

A priori you would take a trajectory of such a system to be a random trajectory, that is x: \Omega \times \mathbb {N} \to X, where \Omega is equipped with some measure, which obeys the law of the system, that is, the distribution of x(\omega ,n+1) conditional on x(\omega ,n), is given by the transition rule of the system, and which is furthermore Markov (if we don't require Markovness, we lose the desirable property that the behavior of a system is entirely determined by the initial condition and the transition rule).

However, in CST, trajectories are chart-type morphisms between systems. There appears to be no good way to make these maps work like the above.

The closest idea is to take a system with state space \Omega \times \mathbb {N}, where the transition ticks up the natural number and resamples the omega to destroy all the information pertaining to x_m for m>n. A trajectory with this clock is a trajectory in the above sense, but we would like to abstract out the choice of a concrete sample space (note also that there is no fixed distribution on \Omega , so eg we can't ask for the distribution of x_0 in the above).

Apart from figuring out the right way to take the "sum" over such \Omega , there is also the question of whether we can pick out the ones which are "good" in the sense that the trajectories they classify are the Markov processes with the right transition probabilities (this is encoded in the above example in the way \omega \in \Omega is resampled).