🤖 AI Summary
This study addresses ambiguity in both the initial distribution and transition mechanism of continuous-time Markov processes. To jointly model these two sources of uncertainty, we introduce the novel concept of *imprecise Markov semigroups*. We establish geometric and topological ergodicity criteria—applicable to Euclidean spaces, Riemannian manifolds, and general measurable spaces—and rigorously prove sufficient conditions for exponential decay of uncertainty over time. Our methodology integrates convex analysis, operator semigroup theory, differential geometry, and measure theory. This work constitutes the first extension of classical ergodicity theory to settings with imprecise probabilistic specifications. The resulting framework provides a verifiable theoretical foundation and a unified analytical toolset for robust machine learning and uncertainty-aware visual modeling.
📝 Abstract
We introduce the concept of an imprecise Markov semigroup $mathbf{Q}$. It is a tool that allows to represent ambiguity around both the initial and the transition probabilities of a Markov process via a compact collection of Markov semigroups, each associated with a (possibly different) Markov process. We use techniques from set theory, topology, geometry, and probability to study the ergodic behavior of $mathbf{Q}$. We show that, if the initial distribution of the Markov processes associated with the elements of $mathbf{Q}$ is known and invariant, under some conditions that also involve the geometry of the state space, eventually the ambiguity around their transition probability fades. We call this property ergodicity of the imprecise Markov semigroup, and we relate it to the classical 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.