  We introduce the concept of imprecise Markov semigroup. It allows us to see
Markov chains and processes with imprecise transition probabilities as (a
collection of diffusion) operators, and thus to unlock techniques from
geometry, functional analysis, and (high dimensional) probability to study
their ergodic behavior. We show that, if the initial distribution of an
imprecise Markov semigroup is known and invariant, under some conditions that
also involve the geometry of the state space, eventually the ambiguity around
the transition probability fades. We call this property ergodicity of the
imprecise Markov semigroup, and we relate it to the classical (Birkhoff's)
notion of ergodicity. We prove ergodicity both when the state space is
Euclidean or a Riemannian manifold, and when it is an arbitrary measurable
space. The importance of our findings for the fields of machine learning and
computer vision is also discussed.
