I will present an observation model that I used, in a recent collaboration with colleagues from Göttingen and Berlin universities, in the inference of large scale stable behaviours for stochastic (Markovian) dynamical systems, from discrete observations of their transitions. Such stable structures are represented in the spectral components of the transfer operator associated with the Markov kernel of the dynamic. The same operator can be constructed directly from the observations, however, the resulting application cannot recover the limit one in a strong enough sense to approximate its spectrum. Instead, we used entropic optimal transport plans in order to regularize the discrete operator and re-introduce, in its definition, the geometric information about the base space, contained in the observed transition pairs. 

I will demonstrate the effectiveness of this construction in the analysis of stochastic dynamical system in the form of a (quantitative) convergence result for the spectral characteristics of induced entropic transfer operator and a non-quantitative one for entropicly regularized transition plans obtained via log-likelyhood inference in a slightly different setting. I will also discuss how an embedding of the particles can be derived from these regularized objects, in order to extend the analysis to trajectories not yet sampled.